Design of a 250 GHz Gyrotron Amplifier
by
Emilio A. Nanni
B.S. Electrical Engineering, University of Missouri - Rolla (2007)
B.S. Physics, University of Missouri - Rolla (2007)
Submitted to the
Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degree of
ARCHIVES
MSSACL
Master of Science
sT
O
at the
/1
OCT
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2010
@ 2010 Massachusetts Institute of Technology. All rights reserved.
/
.
Authoi .........
Department of Electrical Engineering and Computer Science
August 17, 2010
. . . . . . . . . . . . . . . . .
Certified by ..................
. . . . .
......
Richard J. Temkin
Senior Research Scientist, Department of Physics
Thesis Supervisor
A ccepted by .........................
'Professor Terry P. Orlando
Chairman, Committee on Graduate Students
Department of Electrical Engineering and Computer Science
i
E'T INSTI
a5 2
i
2
Design of a 250 GHz Gyrotron Amplifier
by
Emilio A. Nanni
Submitted to the Department of Electrical Engineering and Computer Science
on August 17, 2010, in partial fulfillment of the
requirements for the degree of
Master of Science
Abstract
A design is presented of a 250 GHz, 1 kW gyrotron traveling wave tube (gyro-TWT)
amplifier with gain exceeding 50 dB. Calculations show that the amplifier will operate
at 32 kV, 1 A with a saturated gain of 60 dB, an output power of 1 kW and a gain
bandwidth of 3 GHz. The amplifier uses a novel photonic band gap (PBG) interaction
circuit for stable single mode operation in an overmoded circuit. The design mode is
a TE0 3 -like mode confined by the PBG lattice with no nearby competing modes. The
design of the input coupler, interaction circuit, output coupler and electron gun have
also been completed. The amplifier design accounts for requirements imposed by the
device's intended application of pulsed nuclear magnetic resonance spectroscopy with
pulses as short as several hundred picoseconds. This design will be implemented at a
later date for the construction of the highest frequency, kW power level amplifier based
on vacuum electronics. To gain understanding of short pulse amplification in vacuum
electron devices, an experimental study of sub-nanosecond pulse amplification in a
gyro-TWT has been carried out on an existing experimental setup at 140 GHz. The
gyro-TWT operates with 30 dB of small signal gain in the HE 06 mode of a confocal
waveguide. Picosecond pulses show broadening and transit time delay due to two
distinct effects: the frequency dependence of the group velocity near cutoff and gain
narrowing by the finite gain bandwidth of 1.2 GHz. Experimental results taken over
a wide range of parameters, with pulses as short as 400 ps, show good agreement with
a theoretical model in the small signal gain regime.
Thesis Supervisor: Richard J. Temkin
Title: Senior Research Scientist, Department of Physics
4
Acknowledgments
I would like to thank my advisor, Dr. Richard Temkin, for his guidance and mentorship. He never fails to provide good suggestions and is an invaluable source of
knowledge. I would also like to thank Dr. Michael Shapiro for many useful conversations and for patiently answering all of my questions, and Dr. Jagadishwar Sirigiri
for all his help in the lab and for keeping me on track. I have benefited from working
with all of our group members: Dr. Alan Cook, Dr. Hae Jin Kim, Dr. Colin Joye,
Elizabeth Kowalski, Dr. Roark Marsh, Ivan Mastovsky, Brian Munroe, David Tax
and Antonio Torrezan de Sousa. You have made this process so enjoyable. I want
to thank my instructor for all things DNP, Alexander Barnes. I could never forget
my mentors Dr. Reza Zoughi and Dr. Ronald Bieniek, both from the University of
Missouri-Rolla, who encouraged me to be a better student, scientist and engineer.
On a personal note, I must also thank my parents Antonio and Valeria Nanni who inspired me and provided me with every opportunity. Finally, my fiancee Sarah Wilson,
your love is all the support I will ever need.
Contents
1
2
3
17
Introduction
1.1
M otivation . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
17
1.2
Gyrotron Oscillators and Amplifiers . . . . .
. . . . . . . . . . . .
18
1.3
Background of Gyro-TWTs
. . . . . . . . .
. . . . . . . . . . . .
20
1.4
Current Effort . . . . . . . . . . . . . . . . .
. . . . . . . . . . . .
22
25
Theory of Gyro-devices
2.1
Axial and Azimuthal Bunching of Electrons
. . . . . . .
. . . . . .
25
2.2
Fundamentals of Gyrotron Oscillators . . . . . . . . . . .
. . . . . .
28
2.3
Background . . . .
. . . . . .
30
2.4
Nonlinear Theory of Gyrotron Oscillators . . . . . . . . .
. . . . . .
32
2.4.1
Derivation . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
32
2.4.2
Analysis......... . . .
. . . . . . . . . ..
. . . . . .
37
2.5
Quantum Mechanical Approach for Gyrotron Oscillators
. . . . . .
40
2.6
Theory of Gyrotron Traveling Wave Tubes . . . . . . . .
. . . . . .
43
2.7
Short Pulse Propagation and Amplification . . . . . . . .
. . . . . .
45
2.8
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
52
. . . . . . . . . . . . . . . . . . ..
53
250 GHz Gyro-TWT Design
. . . . . . . . . . . . . . . . .
3.1
Introduction . . . . . . . . . . . .
3.2
Attenuation in Waveguides . . . . . . . . . . . . . . . . . . . . . . . .
53
55
4
3.3
Photonic Band Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3.4
PBG Interaction Circuit . . . . . . . . . . . . . . . . . . . . . . . . .
62
3.5
Linear Growth Rate
. . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3.6
Input Coupler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
3.7
Output Coupler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
3.8
Electron Gun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
3.9
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
Short Pulse Amplification
89
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
4.2
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
4.3
Short Pulse Amplification.... . .
. . . . . . . . . . . . . . . . . .
93
Pulse Broadening Curves . . . . . . . . . . . . . . . . . . . . .
99
4.3.1
4.4
Summary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
List of Figures
. . . . . . ...
2-1
The solution to 2.5 for Ge/Yowp = 10 and -yo = 1.02.
2-2
Cross section of the interaction region along the axis of the DC mag-
27
netic field with (a) a view of the electron beam passing through the
physical structure, (b) the amplitude (c) and phase of the RF electric
field and (d) the DC magnetic field [26].
2-3
. . . . . . . . . . . . . . . .
29
Dispersion diagram for cyclotron resonance and cylindrical waveguide
modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2-4
Power output for vacuum electron and solid state devices [42].
31
2-5
Cross section of the interaction region with black dots representing
. . . .
electrons on their gyration orbit [27]. The waveguide radius is ro, the
Larmor radius is rL, the guiding center radius is Re and EO denotes
the 9 component of the electric field..........
2-6
Contour plot for efficiency of the
1"
.... .
. ...
harmonic interaction between a
cavity with normalized field strength, F, and interaction length, p. . .
2-7
Contour plot for efficiency of the
2
39
The fraction of electron energy that is extracted from the beam, averaged over 32 initial phases, as it passes through the cavity. . . . . . .
2-9
38
"d harmonic interaction between a
cavity with normalized field strength, F, and length, p. . . . . . . . .
2-8
33
40
Stimulated emission of an electron in a magnetic field with relativistic
and nonrelativistic treatment. . . . . . . . . . . . . . . . . . . . . . .
42
3-1
Dispersion relation for a circular waveguide with a 2 mm radius.
3-2
Triangular lattice of metallic rods. The fundamental unit cell is marked
.
54
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
Reciprocal triangular lattice where b' = -,.
The gray area is the
73b
irreducible Brillouin zone. . . . . . . . . . . . . . . . . . . . . . . . .
58
with a dashed line.
3-3
3-4
. .
(a)Master/slave boundaries for the unit cell and (b) the equivalent
HFSS model with the complex magnitude of the electric field of the
lowest order propagating mode for a/b = 0.2,
#1 =
7r/9 and
#1 =
197r/18. 60
3-5
Normalized eigenmode frequencies as a function of ki for a/b = 0.2.
61
3-6
Normalized eigenmode frequencies as a function of ki for a/b = 0.43.
61
3-7
The global band gap plot for a triangular lattice, where a is the spacing
and b is the diameter of the rod. . . . . . . . . . . . . .
3-8
. . . . .
62
The global band gap plot with the operational spacing of a/b = 0.43
is marked with a dashed red line. The TE03 mode for a finite lattice
with three rods removed is marked with a red point. . . . . . . . . . .
3-9
64
The complex magnitude of the electric field for the TE0 3 mode confined
in a circular waveguide. . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3-10 The complex magnitude of the electric field for the TE0 3 -like mode
confined in a PBG composed of a triangular lattice of metal rods. . .
66
3-11 Attenuation as a function of frequency for the TE0 3 -like mode. . . . .
67
3-12 Normalized coupling coefficient comparison for a circular waveguide
and the PBG waveguide. The electron beam radius for the gyro-TWT
w ill be 1.1 mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-13 Gain in dB/cm from linear theory.
. . . . . . . . . . . . . . . . . . .
67
70
3-14 Gain in the device calculated from the linear growth rate and loss per
unit length of the circuit. Reflections are suppressed enough to prevent
oscillations.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
3-15 Gain in dB/cm at 250 GHz as a function of perpendicular velocity
spread calculated from the linear growth rate and MAGY. . . . . . .
72
3-16 Gain in the device calculated from the linear growth rate for a 22 cm
circuit compared to results from MAGY for a 26 cm circuit with 5%
perpendicular velocity spread. . . . . . . . . . . . . . . . . . . . . . .
73
3-17 Dimensions of circuit simulated in MAGY. . . . . . . . . . . . . . . .
73
3-18 Output power as a function of frequency simulated with MAGY using
a 32 kV, 1 A electron beam with 5% v 1 spread. . . . . . . . . . . . .
74
3-19 Output power as a function of the input power for 250 GHz. . . . . .
74
3-20 Band gaps in at triangular lattice. . . . . . . . . . . . . . . . . . . . .
76
3-21 S-parameters for the wraparound input coupler. . . . . . . . . . . . .
76
3-22 Uptaper dimensions simulated and optimized in CASCADE. . . . . .
78
3-23 S1 of the uptaper for the TE0 3 mode. . . . . . . . . . . . . . . . . . .
78
. . . . . . . . . . . . . . . . .
78
3-24 S21 of the uptaper for the TE0 3 mode..
3-25 Gun parameters calculated using equations derived by Baird and Lawson for a 32 kV 2 A beam and 50 degree cathode tilt. The (a) compression, (b) magnetic field at the emitter, (c) emitter width, (d) maximum
electric field and cathode to mod-anode potential, (e) Langmuir space
charge, and (f) cathode to mod-anode spacing, d, and guiding center
spread are plotted versus the mean emitter radius. Black dots mark
the selected design values. . . . . . . . . . . . . . . . . . . . . . . . .
81
3-26 Schematic of the electron gun design with all units in mm unless otherwise m arked.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
3-27 The magnetic field as used to model the electron gun performance.
The guiding center radius is also plotted along the axis of the magnet.
83
3-28 The main magnetic field with the gun coil magnet at 0% and ±25% of
the rated current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
3-29 Electric field distribution for the cathode at -32 kV, mod-anode at
-18kV and anode at ground. . . . . . . . . . . . . . . . . . . . . . . .
84
3-30 Electric potential lines and particle trajectories for the cathode at -32
kV, mod-anode at -18kV and anode at ground. Gun coil at 25% of
rated current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
3-31 Cross sectional CAD drawing of 9.6 Tmagnet. . . . . . . . . . . . . .
86
3-32 Operational parameters for the electron gun at 32 kV. Gun coil at 25%
of rated current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
3-33 Operational parameters for the electron gun at 40 kV. Gun coil at 25%
of rated current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
3-34 Operational parameters for the electron gun at 50 kV. Gun coil at 25%
of rated current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-1
87
Dispersion diagram of HE0 6 gyro-TWT in a confocal waveguide (left)
and cross sectional view (right). The annular electron beam has a 4.2
mm diameter......... . . . . . . . .
4-2
. . . . . . . . . .
. . . . .
Measured (dots) and calculated (solid line) gain-bandwidth curve for
a beam voltage of 28 kV. . . . . . . . . . . . . . . . . . . . . . . . . .
4-3
90
91
Measured (dots) and calculated (solid line) gain-bandwidth curve for
a beam voltage of 35.4 kV. . . . . . . . . . . . . . . . . . . . . . . . .
91
4-4 Measured input and output power at 137.8 GHz for the 35.4 kV operating point. The gain is 23.5±0.34 dB
4-5
. . . . . . . . . . . . . . . . .
Schematic diagram of the picosecond pulse amplification in a quasioptical gyro-TW T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-6
93
Attenuation as a function of frequency in a 10 cm section of WR4
w aveguide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-7
92
95
Measured (dots) and calculated (solid line) transit time delay of 500
ps pulses in (a) 10 cm long WR4 waveguide with a cutoff frequency
of 139.25 GHz and (b) the 28.4 cm gyro-TWT confocal waveguide
structure with a cutoff frequency of 135.0 ± 0.3 GHz. . . . . . . . . .
4-8
Output pulse width as a function of input frequency for 10 cm WR4
waveguide and a constant input pulse width of 500+25 ps. . . . . . .
4-9
96
96
Output pulse width as a function of input frequency for a beam voltage
of 28kV and a fixed input pulse width of 425±25 ps . . . . . . . . . .
98
4-10 Pulse shapes for an input pulse of 580 ps at the 35.4 kV operating
point. The output pulse at (a) 137.3 GHz has a measured width of
1045 ps and (b) at 138.13 GHz has a measured width of 660 ps.
. . .
98
4-11 Measured (dots) and calculated (solid lines) output pulse width after
amplification as a function of input pulse width for 28 kV and (a) 136.94
GHz, (b)137.4 GHz, (c) 137.92 GHz and (d)138.2 GHz. . . . . . . . .
100
4-12 Measured (dots) and calculated (solid lines) output pulse width after amplification as a function of input pulse width for 35.4 kV and
(a)137.3 GHz, (b)138.13 GHz and (c)138.8 GHz. . . . . . . . . . . . .
101
14
List of Tables
21
.....................
1.1
Recent Gyro-TWT Experiments ......
2.1
Illustrative THz Gyrotron Oscillator Experiments . . . . . . . . . . .
31
3.1
Simulation Parameters to Investigate Brillouin Zone . . . . . . . . . .
60
3.2
Design Operating Parameters from Linear Theory . . . . . . . . . . .
69
3.3
Design Operating Parameters from MAGY.. . . . .
. . . . . . .
72
3.4
Electron gun parameters . . . . . . . . . . . . . . . . . . . . . . . . .
80
16
Chapter
Introduction
1.1
Motivation
Novel high power sources in the millimeter and sub-millimeter wave are of great interest due to their potential applications in spectroscopy, communications and radar.
This thesis presents the design of a 250 GHz, 1 kW gyrotron traveling wave tube amplifier for nanosecond pulses with gain exceeding 50 dB. The amplifier will eventually
be used in pulsed dynamic nuclear polarization (DNP) nuclear magnetic resonance
(NMR) and electron paramagnetic resonance (EPR) experiments.
Controlled mi-
crowave irradiation of an NMR sample can be used to transfer the high electron
polarization to the nucleus under study resulting in a dramatic increase in the NMR
signal by factors exceeding 100 [3]. Currently, there are no commercially available
devices that can generate peak powers in the kW region with high gain at 250 GHz.
A gyrotron amplifier was chosen because it offers high gain, broad bandwidth at
these frequencies and allows the use of an overmoded interaction structure resulting
in easier fabrication and reduced thermal effects.
Pulsed DNP NMR requires the amplification of pulse trains with ns times scale
resolution. In principle an amplifier with over 1 GHz of instantaneous bandwidth
should be able to meet this requirement. However, waveguide dispersion and gain
dispersion remain a concern for successful operation. In light of these concerns a study
of pulse amplification in a 140 GHz gyrotron traveling wave tube (gyro-TWT) was
performed. The 140 GHz gyro-TWT used in the experiment was developed at MIT for
use in pulsed DNP. This study was the first investigation of sub-ns pulse amplification
in a gyro-TWT, and yielded the first dispersion free amplification of sub-ns Gaussian
pulses.
The effects of waveguide dispersion and gain dispersion were thoroughly
investigated in the linear gain regime of the amplifier. A theory was developed to
describe the effects of waveguide and gain dispersion on pulse amplification in gyroTWTs, in excellent agreement with experimental results.
1.2
Gyrotron Oscillators and Amplifiers
The generation of millimeter and sub-millimeter wavelength radiation at high power
has proved to be a significant challenge. Solid state devices have never been considered
as a possibility for high power generation at such high frequencies (hundreds of GHz)
due to scalability and efficiency issues. Classical microwave tubes, e.g. klystrons and
traveling wave tubes, can produce high power (kW) electromagnetic radiation up to
100 GHz, but these slow wave devices require physical structures in the interaction
cavity that are smaller than the wavelength of operation. This small element size
produces difficulties with thermal damage and manufacturing of the interaction cavity. Gyrotrons are a form of electron cyclotron maser capable of producing kilowatts
to megawatts of output power in the microwave, millimeter wave and terahertz bands
[18, 12, 22, 36]. These devices operate by the resonant interaction between the eigenmodes of an interaction cavity, typically cylindrical, and a mildly relativistic electron
beam that is gyrating in a constant axial magnetic field. The most basic configuration
of a gyrotron consists of a magnetron injection gun (MIG) that launches an annular
electron beam into the hollow bore of a solenoidal, often superconducting, magnet.
The orientation of the DC electric field that extracts the electron beam from the cathode produces a beam that has both a perpendicular and parallel velocity component
to the axial field produced by the solenoidal magnet. As the electron beam travels
into the central bore, it undergoes adiabatic compression that increases its orbital
momentum. The beam enters a metallic cavity that has an eigen-mode resonance
that is close to a harmonic of the frequency at which the electron gyrates around the
magnetic field line. The electron beam surrenders some of its kinetic energy to the
electromagnetic mode through stimulated emission. The electrons exit the cavity and
are deposited on a metallic collector. In its simplest configuration, the collector can
also hold an axial dielectric window that couples the electromagnetic radiation out of
the device. However, it is often convenient to convert the wave into a Gaussian beam
that can be efficiently extracted in a transverse direction using mirrors.
In recent years, gyrotron traveling wave tube (TWT) amplifiers have demonstrated
high output power levels with significant gain bandwidths [13, 11, 7]. A gyrotronTWT amplifier works on the same fundamental principles as a gyrotron oscillator for
the extraction of energy from an electron beam. However, the amplifier is operated
under conditions that suppress self-start oscillations, including backward wave oscillations (BWOs) that could disrupt the operation of the device. Amplification is achieved
in a gyro-TWT by a convective instability supported by a mildly relativistic, annular,
gyrating electron beam and a transverse electric mode in a waveguide immersed in
a strong static axial magnetic field (BO). The grazing intersection between the dispersion relation of the cyclotron resonance mode and a transverse electric waveguide
mode near the waveguide cutoff results in high gain and moderate bandwidth. The
beam mode dispersion relation is given by
w - sG/7 - kzvz
0
(1.1)
and the waveguide mode dispersion relation is expressed as
w2 - k2zc 2 - k2Ic 2 = 0
(1.2)
where w is the frequency of the wave, Q = eBo/me is the non-relativistic cyclotron
frequency of the gyrating electrons, e and me are, respectively, the charge and the
rest mass of the electron, -y is the relativistic mass factor, s=1 is the cyclotron harmonic number, k, and ki are the longitudinal and transverse propagation constants,
respectively, of the waveguide mode, vz is the axial velocity of the electrons, and c is
the velocity of light. This requires the design of an interaction circuit that supports
the propagation of a mode at the correct frequency while appearing lossy to frequencies and modes outside the region of interest. This is usually performed by inserting
lossy ceramics, severs and selecting geometries with favorable dispersion relations
[29]. Further complications arise when considering the design of an amplifier at 250
GHz because the operational mode of the amplifier cannot be a fundamental mode
due to compression restrictions of the electron beam. In a gyro-TWT, the signal of
interest must be efficiently coupled into the cavity at the beginning of the interaction
region. The wave propagates along the amplifier circuit while extracting energy from
the electron beam until it is removed from the device with an output coupler in a
similar fashion to the oscillator.
1.3
Background of Gyro-TWTs
Gyro-TWT amplifiers were first studied in the early 1970s, mainly at the Naval Research Laboratory (NRL). These devices were in the X and Ku bands and operated in
the fundamental waveguide mode [4, 39]. These early studies showed how the promise
of high gain and high output power in gyro-TWTs was limited by the strong forward
and backward wave gyrotron oscillations near the waveguide cut-off and interaction
at higher harmonics.
More recently, a scheme of distributed loading was proposed by the research group
at the National Tsing Hua University (NTSU) in Taiwan to suppress the gyrotron
backward wave oscillations (gyro-BWO) and forward wave gyrotron oscillations near
the waveguide cut-off. A proof-of-principle device demonstrated over 70 dB of gain at
35 GHz [13]. This device operated in the fundamental TE11 waveguide mode. Subsequently, a TEOi high average power device based on this concept was demonstrated
at NRL [35] and a TEOi device was demonstrated at 94 GHz by Communication and
Power Industries (CPI) [8]. A comparison of recent gyro-TWT experiments is shown
in Table 1.1.
Table 1.1: Recent Gyro-TWT Experiments
NTHU[13]
CPI[8]
MIT (2003)[42]
MIT (2008)[29]
Freq (GHz)
35
95
140
140
Gain (dB)
70
43
29
34
Pout (kW)
93
1.5
30
0.8
Voltage (kV)
100
30
65
37
Current (A)
3.5
1.8
7
2.7
However, operation at fundamental modes is not feasible at high frequencies such
as 250 GHz where the radius of the interaction structure would be of the order of
a fraction of a millimeter. This results in high ohmic losses and presents significant
challenge in transporting an electron beam over long distances (~20 cm) without
causing beam interception on the waveguide walls. In prior research at MIT, gyroTWTs with high power (30 kW) and low power (0.8 kW) have been successfully
demonstrated at 140 GHz [44, 29].
These experiments used a novel quasioptical,
overmoded interaction structure which allows for single mode operation in a higher
order mode. The interaction circuit consists of a confocal waveguide with severs for
additional suppression of oscillations. Such a mode-selective open waveguide imparts
high diffractive losses to the lower order modes which tend to interact with the beam
more strongly.
This selective loading of the lower order modes allows for stable
operation in a higher order mode. Though mode selective, the confocal waveguide
has an azimuthally asymmetric field profile which reduces its interaction efficiency
with the annular beam produced by the MIG that is typically used in gyrotrons.
One important application of millimeter waves is in spectroscopy, where coherent
pulses on a sub-nanosecond or picosecond time scale are needed for optical pumping
of molecular states. The pulses must be shorter than the relaxation time, typically
requiring sub-nanosecond (or picosecond scale) pulse lengths [6].
Sub-nanosecond
microwave pulses have been demonstrated in a vacuum electron device by superradiance [21], but such pulses cannot be used for spectroscopy. Sub-nanosecond pulses
must contain a spectral bandwidth exceeding the transform limit of 1 GHz. In the
conventional microwave bands at frequencies of one to several GHz, the required gain
bandwidth to amplify such picosecond pulses is generally not available, since the GHz
bandwidth is a large fraction of the carrier frequency. In recent years, high power,
wideband amplifiers in the millimeter wave band have been developed that are suitable for amplifying picosecond pulses. For example, a form of klystron called an
extended interaction klystron has been developed at 95 GHz with a gain bandwidth
of about 1 GHz. This amplifier has been used to successfully amplify 1 kW output
pulses as short as 800 picoseconds [9].
A gyro-TWT at 95 GHz has been demon-
strated with a gain bandwidth of 6.5 GHz at an output power level of 2 kW [7]. This
gyro-TWT could in principle be used to amplify a 150 ps pulse. To our knowledge,
amplification of picosecond pulses has not been tested with these or other powerful
wideband gyro-amplifiers. Because of the possibility of distortion in amplification of
picosecond pulses, detailed studies are needed of the amplification process in such
devices.
1.4
Current Effort
A novel interaction structure that is highly mode selective and has strong coupling
with the electron beam resulting in higher interaction efficiency will be presented.
The circuit is based on a photonic band gap (PBG) structure made of a two dimensional triangular lattice of metal rods. The dimensions of the lattice are tuned so
that it acts as a perfect reflector in a narrow band of frequencies around the operating mode. A defect is created in the lattice by removing some rods, allowing a
higher order mode to be confined with high a quality factor (Q). Other modes that
can exist in the defect either at higher or lower frequencies suffer significant losses
because of the partially transparent lattice. This amplifier will ultimately be applied
to coherent molecular spectroscopy. Therefore, the bandwidth of the amplifier has
been an important consideration to insure the ability of the gyro-TWT to faithfully
amplify very short pulses without distortion, since it is important to the application.
The input coupler, output coupler and electron gun designs for this experiment will
be presented.
We report a detailed study of amplification of pulses as short as 400 ps in a 1
kW, 140 GHz gyro-TWT with a gain bandwidth exceeding 1 GHz. The gyro-TWT
operates with 30 dB of small signal gain near 140 GHz in the HE 06 mode of a confocal
waveguide.
Picosecond pulses show broadening and transit time delay due to two
distinct effects: the frequency dependence of the group velocity near cutoff and gain
narrowing by the finite gain bandwidth of 1.2 GHz. A comparison between theory
and experiment in the dispersion of picosecond pulses in a vacuum electron device
amplifier was performed by driving the amplifier with pulses as short as 400 ps, which
contain frequencies over a 2.5 GHz bandwidth, about twice the amplifier's inherent
gain bandwidth. Experimental results show good agreement with a theoretical model
in the small signal gain regime. These results show that in order to limit the pulse
broadening effect in gyro-amplifiers, it is crucial to both choose an operating frequency
at least several percent above the cutoff of the waveguide circuit and operate at the
center of the gain spectrum with sufficient gain bandwidth.
24
Chapter2
Theory of Gyro-devices
2.1
Axial and Azimuthal Bunching of Electrons
An electron beam propagating along a magnetic field line is subject to a variety of
instabilities. Of particular importance are the axial and azimuthal bunching mechanisms of the electromagnetic electron cyclotron instability. The electron cyclotron
maser instability is driven by the azimuthal bunching mechanism and requires a relativistic treatment. The Weibel-type instability is driven by axial bunching and does
not require relativistic treatment. Both mechanisms exist and compete against each
other when an external magnetic field is present [15].
Assume an infinite homogeneous magnetic field Bos with the electron distribution
given by
fo = 6(p_ - pio)6(p2)/27rp1
(2.1)
where pi is the transverse momentum, p, is the axial momentum and pLo is a constant. This distribution represents streaming electrons in the moving frame. This
monoenergetic beam would be the ideal case for an experiment designed to extract
energy from the electrons, and it approximates the electron beam produced by high
quality electron guns.
If we perturb the electron beam with an electromagnetic wave, the system will
require the linearized relativistic Vlasov equation,
a
e
a
a fi=e
fi+v---f1--vxBoat
8x
c
89p
1
E1+-vxB1
c
a fo
ap
(2.2)
and the combined Maxwell's equations,
1 82
V x V x Ei = -- 2
c at
2
E -
47r a
c2
J
(2.3)
2t
to fully describe the physics of the interaction. A subscript 0 or 1 represent the initial
or perturbed quantity, respectively. The electron distribution is f, E is the electric
field, B is the magnetic field, and the current J is defined as
J = -e Ifivd3P.
(2.4)
Simplifying the above equations and performing perturbative analysis for a wave that
propagates as exp(-iwt + ikzz), we obtain the relativistic dispersion relation [15]:
2C2
= -rw
kzpz
xKW -)/
pidpj
pi''
afo +
dpz
kz 2 afo (-k"1--- (yo-
where w2 = 47rne2 /m, Qe = eBo/mc, and 7
kzpz/m - e) 1
(1 + p2/m
expression is for any generic distribution of the electrons.
2c 2
+ p /m 2c2 ) 1/ 2 . This
However, we can sim-
plify further by considering the monoenergetic case from Equation (2.1) that is most
relevant to practical devices. This reduces the dispersion relation to
2
2
w2 - k c
W
_ "2
Z
- , /7
yo
where -yo = (1 + piO/m 2c 2) 1 /2 and
#10o
#i2
-kc2
2(uo - Ge/70)2(25
(2.5)
= pio/yomc. To extract some physical inter-
pretation from Equation (2.5) one should make note that the last term on the RHS is
where the axial and azimuthal bunching mechanism reside. The azimuthal and axial
bunching are represented by the w 2 and c2 k2 terms respectively. Furthermore, the
........................................................................................................
denominator (w - Ge/yo) imposes a requirement that the frequency of oscillation for
the wave must meet the condition w ~ Ge/Yo in order to produce a large instability.
This can be understood as a requirement for synchronous behavior between the electrons and the electromagnetic wave. Additionally, if the phase velocity of the wave is
equal to the speed of light there is no instability present, defining an inflection point
between an instability dominated by axial and azimuthal bunching. If
W2/k
>
(2.6)
C2 ,
the phase velocity of the wave is greater than the speed of light and azimuthal bunching dominates. If
w 2 /k
(2.7)
<c 2 ,
the phase velocity of the wave is less than the speed of light and axial bunching
dominates. In Figure 2-1 the solution to Equation (2.5) is plotted for Ge//yowp = 10
and yo = 1.02. We note that the instability is only present when the frequency of
the fast and slow wave meet the condition w/w, = 10 as required for synchronous
behavior. The instability for the fast and slow wave are only present when Equations
(2.6) and (2.7) are satisfied.
Fast Waveo
r
Fast Wave 1On.
Slow Wave or
- -Slow Wave 100io.
C.
5
10
15
kz c/o
20
25
p
Figure 2-1: The solution to 2.5 for Qe/yowp = 10 and yo = 1.02.
..
Gyrotrons and gyro-TWTs are both fast wave devices that rely electron cyclotron
maser instability, driven by the azimuthal bunching mechanism. Due to this bunching
mechanism a relativistic treatment is required to explain these devices properly.
2.2
Fundamentals of Gyrotron Oscillators
As mentioned in the previous section, electron cyclotron resonance masers (gyrotrons)
are fast wave devices, i.e. the phase velocity of the wave is greater than the speed of
light. They are capable of producing high average power in the microwave, millimeter
wave and sub-millimeter wave range. The most basic configuration of a gyrotron
can be seen in Figure 2-2. The magnetron injection gun that launches an annular
electron beam would be located to the left of Figure 2-2(a). The axial magnetic field
is produced by a solenoidal magnet than can either be a pulsed magnet or a DC
superconducting magnet. As the electron beam travels through the central bore, it
enters a metallic cavity that has an eigen-mode resonance that is close in frequency to
a harmonic of the frequency at which the electron gyrates around the magnetic field
line. The electron beam surrenders some of its kinetic energy to the electromagnetic
mode through stimulated emission. In order for the electron to interact with the
electromagnetic mode in the cavity, the relativistic electron frequency must be close
to the frequency of oscillation for the mode [14]. The relativistic cyclotron frequency
is
Qrei =
eB 0
'Yme
(2.8)
where B 0 is the DC axial magnetic field, 'y is the relativistic mass factor, me is the
electron mass, e is the charge of the electron. In a cylindrical cavity the dispersion
relation for TE modes is
-
-
z =0,
(2.9)
where k = wJpoo is the wave vector in free space, kt = vmn/rw is the transverse
propagation constant, vmn is the root of a Bessel function and kz is the axial propagation constant. It can be shown [5] that the Doppler shifted resonance for the electron
|+- L--
+|.
--
r -beam
(b)
(c)
(d)
Figure 2-2: Cross section of the interaction region along the axis of the DC magnetic
field with (a) a view of the electron beam passing through the physical structure, (b)
the amplitude (c) and phase of the RF electric field and (d) the DC magnetic field
[26].
becomes
(
- kzvz - n9rel = 0,
(2.10)
and when the frequency of the mode and the resonance intersect, as seen in Figure 2-3,
oscillation and stimulated emission can occur. The blue and red line in Figure 2-3 are
the electron beam dispersion relations for n = 1 and n = 2, respectively. The blue dot
is a fundamental forward wave oscillation, the red dot is a second harmonic forward
wave oscillation and the green dot is a fundamental backward wave oscillation.
600
1st Harmonic
550 2nd Harmonic
500.......
Waveguide
Mode.
50 0
.....
...........
.......
-..
450
400
350
LL
300
250-
150-
-2000
0
2000
4000
6000
8000
kZ
Figure 2-3: Dispersion diagram for cyclotron resonance and cylindrical waveguide
modes.
2.3
Background
Gyrotrons were initially proposed due to the cyclotron resonance instability that was
discovered independently in the late 1950s by R. Twiss [48], J. Schneider [40] and A.
Gapanov [19]. This led to the invention of the gyrotron in a similar configuration
to the one described in Section 2.2 by Gapanov et al. [20] with 190 W CW power
at 1.2 cm wavelength. This was followed by the design of gyrotrons in the 1970s
that could produce several kW of power at frequencies up to 300 GHz by Zaystev et
al. [51]. To minimize the thermal load, the production of pulse gyrotron oscillators
with several hundred kW of power and high frequencies was completed in the 1980s
[47].
In the past two decades the focus has shifted from simply achieving higher
frequency devices to providing useful millimeter and sub-millimeter wave sources for
Electron Cyclotron Resonance Heating (ECRH) of plasmas [46], Nuclear Magnetic
Resonance Dynamic Nuclear Polarization (NMR-DNP) and Electron Paramagnetic
.
...............
:::_: ::::..::
Table 2.1: Illustrative THz Gyrotron Oscillator Experiments
Source
Year
MIT[27]
IAP[51]
Fukui[28]
2004
1973
1998
Freq.
(GHz)
460
326
301
Cvcl.
harm.
2
2
1
Mode
(TEnp)
TEO,6 ,1
TE 2 ,3 ,1
TEo, 3 ,1
V
(kV)
12.4
27.9
14
I
(A)
0.13
0.9
0.08
P
(W)
8
1500
17
Resonance (EPR) experiments [3]. This changed the focus to providing reliability,
efficient power conversion and delivery, larger bandwidth and signal stability. A new
class of device called the gyro-TWT also emerged from the discovery of the cyclotron
resonance instability [14] that operates on similar principles, described in Section 2.6
of this paper. Table 2.1 shows some leading CW and pulse gyrotron experiments in
terms of frequency of operation, power output, and bandwidth. In Figure 2-4, the
achieved single device power output for solid state and vacuum electron devices is
plotted. The THz gap is clearly visible with conventional gyrotrons having a clear
supremacy for the 100-500 GHz range.
107
106
105
104
103
102
10
1
10-1
10-2
.1
1
10
100
1,000
10,000
100,000
Frequency (GHz)
Figure 2-4: Power output for vacuum electron and solid state devices [42].
2.4
Nonlinear Theory of Gyrotron Oscillators
2.4.1
Derivation
To understand the operation of a gyrotron oscillator, we must describe the interaction
between an energetic electron and the electromagnetic mode confined in the resonator
or interaction cavity. Like all vacuum electron devices, gyrotrons extract kinetic
energy from the electron beam as it passes through the interaction cavity. However, a
gyrotron does not extract any energy from the axial component of the velocity, only
from the transverse direction.
This energy extraction is made possible because an electron traveling in a circular
path around a DC magnetic field will feel an accelerating or decelerating force from
the oscillating electric field that can deposit or extract energy from the electron. The
initial change in energy that results from the applied force causes more energetic
electrons to rotate slower and less energetic electrons to rotate faster, because of the
change in y and Qrei. The different rotational frequencies can produce phase bunched
electrons that act coherently to deposit energy into the electromagnetic mode of the
cavity. This process can be described by the pendulum equations that relate the
change in energy and momentum for the electron to the electric and magnetic fields
that are present. These equations of motion are
at
8p
at
= -ev - E)
(2.11)
e
= -eE - -v
c' x B,
(2.12)
where the electron energy is E, the momentum p, the RF electric field is E, the DC
magnetic field is B and the velocity of the electron is v. We are able to ignore the effect
of the oscillating magnetic field on the electron because the gyration of the electron
is dominated by the axial magnetic field B = Boz. The instantaneous kinetic energy
and momentum of a relativistic electron are E = ymec 2 and |pl = -yfmec respectively,
where y = (1
-
) A- and # = v/c. The I symbolizes the component that is
Figure 2-5: Cross section of the interaction region with black dots representing electrons on their gyration orbit [27]. The waveguide radius is ro, the Larmor radius is
rL, the guiding center radius is Re and EO denotes the 0 component of the electric
field.
transverse to the beam and the || represents the axial or z component. It will be
useful to convert these equations into normalized variables and to solve the standard
and practical case of a cylindrical resonator. The normalized variables will relate to
relevant design constraints to be discussed later. To begin this process, we convert
to a relative energy for the electron w = 1 through O
=
(1-y/70) =(1/
y'yo which relates to the differential
yomec2)=.
It is also convenient to convert to
a normalized axial position Z = wz/3 11oc which relates to the differential through
3 i and
$
= woi 1/
11oL.
The values of 31io and -yo are defined at the beginning
of the interaction region, and w is the angular frequency of the electromagnetic wave.
Applying these conditions to Equation (2.11) yields
p-.
Z
2
(me c) w
TYo/3
pE
(2.13)
In order to distinguish the magnitude and phase of the transverse momentum
of the electron and electric field, we can write these quantities in complex notation,
P = Px+iPy = P+ = P+Ie" and E = Ex+iEy = E+ = jE+je(t+). The angular term
a for the electron's momentum is defined at the center of gyration for the electron
which is at a radius Re as seen in Figure 2-5. Now we can express the energy in terms
of the real part of our complex variables
9m
BZ
e
#|i
2
(mec) w 'yo
=
Re(p*E+).
(2.14)
11
By taking equation (2.12) and considering the complex transverse momentum |p+|ea
we find that
. Dividing by et a and solving for the imaginary
e"aI+I
ia=
+ |P+i
part provides an equation that describes the change in phase with position for the
momentum of the electron
-
-
B9Z
0Ilo
11
Im(p*E+),
(2.15)
W#|||P+|I
where wc = eBo/mecy is the cyclotron resonant frequency. The traditional configuration for a gyrotron consists of a cylindrical interaction region with a down-taper
leading towards the electron gun and an up-taper facing the electron beam collector
and the output window for the microwave signal. With this resonant configuration,
the dominant contribution to the Q-factor will be the diffractive
diffractive
Q is
Q
=
QD.
This
large enough that we can write a fixed expression for the field dis-
tribution f(z) = e-(kiiz)2 which is a valid assumption for our configuration.
The
electric field in the transverse direction is identical to the TEmp modes for a cylindrical waveguide along with a variation in the amplitude in the z direction. The fields
are
E
ER =
E
=
(ERR + Egoo)ei(wt++O),
i(m/ki R)Eof(z)J(kR)e-imtJo,
=
Eo f (z) J (kIR)ei"
e
,
where k 1 = vmp/ro and w ~ nwe. These field expressions are written in the coordinate
frame (R,
0
) which is centered in the middle of the solenoid. As stated previously,
the complex expression for momentum considers the phase of the electron as it rotates
about its center of gyration. Therefore, we also need our field expression to be centered
around the same point. Haldar and Beck were able to convert from a reference frame
in the center of the cavity to the center of the orbit for the electron using Graf's
formula for Bessel functions [23]. This produces a series solution, but only one term
will interact in a coherent fashion with the nth order resonance. This allows us to
express the fields as
E = (Ernr +En#)ei(wt++),
Ern = i(n/kir)Eof(z)Jm±n(kiRe)Ja(k _L r)e-imoe-in(4-*o),
Eon = Eof (z) Jm±n(kiRe)J' (kir)e-imoe-in(*-*o)
In order to evaluate Equations (2.14) and (2.15) we need the complex expression
E+ = Ex + iEY = Re(Ernei(wt+))ei + Re(Eonei(wt+))iei*. Which gives
E+ = -Er sin[wt - n# +
where E,
# - (m - n)O±e
ErnI and EO =En
+ EO cos[wt - n# + V#- (m - n)#o]ieiO,
If we set 6 = wt - n#, the complex expression
becomes
E+= -E,
since
#
sin[O + i# - (m - n)#o]eio + EO cos[O +
4
- (m - n)#o]ie'O
is the angular position of the electron from its gyrocenter, it is related to
the angular phase of its momentum by a =
#
+ r/2, because the position vector is
orthogonal to the velocity vector for the electron. This changes the field expression
to
E+ = Er sin[6 +
4-
(m - n)#o]ie'" + E4 cos[O + V' - (m - n)O]ea,
and allows us to evaluate Equations (2.14) and (2.15) quite simply
aw
a
e
=
8Z
e
p'E
-oec cos[O +
4-
(m - n)#Oo],
(2.16)
-O- = 6o- w -
az
where 60
=
,
yomecw
1 - wco/w, wO
=
p1
Er sin[O +@
-
(m - n)$o],
(2.17)
)
eB/mecyo and p'. = (|p+|/7omec)
=
(0
- 2w
+
w2)1/2. The value of 60 is equivalent to the detuning of the magnetic field as it relates
to the ratio of the initial cyclotron resonance frequency and the frequency of the
electromagnetic wave. Conveniently, we can set V) - (m - n)O-
r/2 because there
is no axial bunching, and the electrons at the input have a random phase about their
gyrocenter, which further reduces our equations to
aZ- = nBop'sin,
D Og - w aZ
Bo
(2.18)
cos .
p'
(2.19)
It is interesting to note that the radial electric field changes the energy of the electron
and the azimuthal electric field affects the phase. In both cases, this affects the phase
bunching of the electrons. It is convenient to rescale our normalized variables by
2w
2
-
Yo1#io
70'
2_ Z = 7r)3
,
and normalize our fields with
Eo ,,4n-1
F o _"L
n
i
min(klRe)-
It is also possible to make the approximation
7-B1L-FOL~
-p
k'r kr nI
nfl 1 0 (1 ~22w - w 2))1/2 ="1
These renormalizations and the approximation alter equations (2.18) and (2.19) to
the form
=2
2"n!
nnn-f1
Ff(() -J' (np')sin ,
O
(2.20)
6
= A --
-n
f
2'n! \
nn1
Ff ((
#1(1 - /3iou/2)
2
J,(pl)
cos 0,
(2.21)
where A = 260,310. We allow the normalization F for the field to be very complicated
because under the condition that
#2 0/2 <
1, then p'
(120 -2w)1/2=O(I
_ 1)1/ 2 .
This condition permits a small argument expansion on the remaining Bessel function
J2'(X)
x"-1. If we apply this expansion we can rewrite our differential equations
as
- = 2Ff(()(1
a(
06_
a(
-
= A - u - nFf(()(1
-
U)/2-1 cos
Finally, we can define our cavity length L by k
file from f(z)
=
e-kIz) 2 to
f(()
e-(2c/)
2
(2.22)
u)n/2 sin 0,
, where
0.
(2.23)
2/L and rewrite the axial field pro[.
We have constructed
the coupled differential nonlinear equations of motion for a gyrating electron interacting with an electromagnetic field. These equations which describe the normalized
transferred energy and phase of the electron are only functions of three parameters:
the normalized field strength F, cavity length p and detuning of the magnetic field
A. However, Equations (2.22) and (2.23) only track the amount of energy that a
single electron surrenders to the electromagnetic mode. Therefore, we need to verify
that for various input boundary conditions the electrons will act coherently. The
energy spread for the electrons at the input is very small allowing the electrons to
have the same gyration frequency, but the phase around the gyrocenter is completely
random (00 = 0 -- 27r). To calculate the actual efficiency of energy conversion for
the device, we have to make sure that the stimulated emission is coherent for all of
the electrons. This efficiency can be calculated by averaging over initial boundary
conditions r_1 =< u((Out) >00.
2.4.2
Analysis
The efficiency analysis for the conversion of energy from kinetic electron energy to
electromagnetic radiation was performed by using a fourth-order Runge-Kutta algo-
009
-
008
--
0.07
00
5
10
1
20
25
Figure 2-6: Contour plot for efficiency of the 1 s' harmonic interaction between a
cavity with normalized field strength, F, and interaction length, p.
rithm and averaging the results for 32 electrons with uniform spacing in initial phase.
For clarity the results are shown as contour plots with variables F and yt and optimized for A. These plots are shown in this form because A is often the easiest
parameter to vary as the magnet can be charged to the desired strength. The spatial
limits for the calculation were set at ( = -V5t/2 to V3p/2 and this is the commonly
accepted cutoff point for the electric field of a tapered gyrotron oscillator.
In Figure 2-6, the contour plot for efficiency as a function of F and y is shown for
the first harmonic, and Figure 2-7 shows the contour plot for the second harmonic.
It is possible that there are other peaks in efficiency for larger values of F and P, but
it is often desirable to operate at the smallest values of F and p. This is because
the longer the cavity, the more expensive the solenoidal magnet, and the larger the
0.22 -
-
0.2 0
76
0.18 -0
0.16 L-0.14 -06
0.12-
0.1
0..5
0.16-0.08
0.5
0.06
0.6
0-
5
10
15
20
-
26
Figure 2-7: Contour plot for efficiency of the 2 ,d harmonic interaction between a
cavity with normalized field strength, F, and length, p.
F, the greater current the electron gun must provide. Through power conservation,
one can show that F = riI, where I is the normalized current. This is the balance
equation that specifies the current the electron gun must supply to the gyrotron. A
direct expression for the actual current can be found, but that is beyond the scope of
this thesis.
It is also possible to show how the average efficiency changes as the electrons
progress through the cavity, as shown in Figure 2-8. It should be noted that the peak
efficiencies for the first harmonic rL = 0.72 and the second harmonic r/ 2 = 0.71 are
in excellent agreement with published values [17, 37, 50].
S- 0.4 -
0.3-
0.2-
0.1 -
-15
-10
-5
0
5
10
15
|L
Figure 2-8: The fraction of electron energy that is extracted from the beam, averaged
over 32 initial phases, as it passes through the cavity.
2.5
Quantum Mechanical Approach for Gyrotron
Oscillators
In order to verify that stimulated emission can occur for gyrating electrons, we must
consider their behavior on a quantum mechanical level. Landau showed [33, 27]
that an electron moving perpendicularly to the magnetic field is a simple harmonic
oscillator with energy levels W = (i + 1)hwo. With the relativistic treatment of the
relativistic Schrddinger equation [40], the result is altered to
Wi = moc 2 [1
+ 2(i +
2
)hwo/moc2i1/ 2
40
-
moc2.
(2.24)
Therefore, with some small approximations, the transition between two states i + 1
and i will emit a photon of frequency
wi,i+1 = (Wi+1 - Wi)/h
=
3
2
1
+[1 + (i + -)hwo/moc
2
2
moc 2 /h[[l + (i + -)hwo/moc
12
+ -(i2 + 3i + 9/4)(hwo/moc 2) 2]
2
2
12
+ -(i2 + i + 1/4)(hwo/moc 2) 2 ]]),
2
(2.25)
wi,i+1 = wo(1 - ihwo/moc 2 ).
If our system of Ni electrons is placed in a alternating electric field it will undergo
transitions wi,i+ 1 and wi,i_ 1 . This will produce a net power transfer of
Pi = Nih(wi,i+1mi,i+1 - wi,i_1wi,i_1).
(2.26)
We need to compute the transition probability wi,i+1 in order to find an expression
for power transfer. The transition probability is given as wi,i+1 = h-2E2(Pii+1)2g(W)
and pi,i+1 = e[(i + 1)h/2mowo]1 / 2 [40]. If we assume a small collision frequency 1/T
the response can be assumed as Lorentzian where g(w) =r/[1 + (wi,i+1
2
- w) 2T ].
Solving for the transfer of power, we find
- ia)(i
= <D= 1 (-ia)(i-I-)
i/[Ni(ep2 Trmo)(E 2 /2)~(1
1+ w~+
(1 - (i(i - 1)a)i'
)2)
+ 1)
2
w) T
2
+
-1
+
(wi,i_ 1
-
(2.27)
)2
where <D is the normalized power transfer, a = hwo/moc 2 . It is convenient to define
a variable x =(wi,i+l -
W)T
and manipulate the expression (neglecting higher order
terms in a) until the normalized power transfer becomes
1
1+x2
where W is the kinetic energy,
Q=
WX
2 Qmoc2(1+x2)2(
woT.
Equation (2.28) is plotted in Figure 2-9 for two cases where we have a relativistic
electron and a nonrelativistic one. Several major observations can be made from this
plot, most importantly that only in the relativistic case, where the energy levels of
. ...............................
..........
....................
. ...
..................
.............
. ................
..........................
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
Figure 2-9: Stimulated emission of an electron in a magnetic field with relativistic
and nonrelativistic treatment.
the gyrating electron become a function of the energy of the electron, can stimulated
emission produce an energy gain for the electromagnetic field. Secondly, the frequency
of the electromagnetic wave that we stimulate the system with must be slightly higher
than the gyrofrequency of the electron, and as the energy of the electron decreases
its gyrofrequency will increase until it begins absorbing energy. There is one other
major finding that should be noted. Unlike a laser system where the excited state
provides only one photon when it undergoes stimulated emission, a gyrotron that
has very energetic electrons has a broader emission line width and can emit multiple
photons as they pass through the cavity. This greatly increases the output power
that can be achieved while maintaining an efficient system. Many of these findings
are similar to the results observed with the electromagnetic treatment derived in
Section 2.4. However, getting absolute efficiency values and design parameters is
much more difficult with the quantum mechanical model and forces us to rely on the
nonlinear single particle treatment in most cases.
2.6
Theory of Gyrotron Traveling Wave Tubes
Similar to the method described previously for gyrotron oscillators, tracking single
particle behavior and averaging over a group of particles with a uniform phase distribution leads to a set of self-consistent non-linear equations that can describe the
amplification process in a gyro-TWT. Excluding space charge effects, this method
developed by Yulpatov [49] and later generalized by Nusinovich et al. [38] can also
be simplified to produce a linear growth rate that is relatively easy to calculate. The
equations of motion for an electron and the field amplitude are given by:
Bi;
-2
az'
8O
oBz'
1
- w)s/2
(1
1 - bw
-{~
A +tmw+
- 1= b
1 - bw
12
=I'
Oz'
2J
0
Re F'e-i
F
s
____
(1 - W)1-s/2
2
(1 _
1-bw
(2.29)
Im Fe-j
s/
J' (0 e -
o
(2.30)
(-0
e
do
(2.31)
where w is the normalized electron energy, 6 is the normalized electron phase,
F is the normalized field amplitude, I6 is the normalized current, z' = kz is the
normalized position, k is the wavenumber, s is the cyclotron harmonic and we define
the normalized parameters
b
-h#t|o
2/3zo(1 - hlQzo)
2
32
fo 1 - h
S
2#3zo 1 - h#20
=
1- s
/#zo (
h =
- h3zo)
7
eBo
meow
kz/k.
Additionally, we note that kz is the axial wavenumber, 32o = 2-, vzo is the initial
axial electron velocity, 3 to =
C,
vto is the initial transverse electron velocity, -yis the
relativistic mass factor, e is the electron charge, meo is the electron mass and Bo is
DC axial the magnetic field. In order to relate the normalized field amplitude to the
waveguide modes
E
Re [AS(rt)e
=
-kzz)1
(2.32)
I
(2.33)
H = Re [AH S(rt)ew
we need the relationship
(Kp'to
1
F = poc1 - hf3zo
3
K 70o/to/
2s(s - 1)!
p
A' Ls(X, Y)
(2.34)
20
/
eA
meoCW
where , = kt/k, kt is the transverse wavenumber p'
(2.35)
- P= to
is the initial transverse
momentum. The normalized electron energy and phase are defined as
_
2(1 - hzo) yo- y
,3t2
(2.36)
O = sO - (wt - kzz)
(2.37)
e
(2.38)
=JZ
where p, is the axial momentum and
# defines
the x and y momentum of the particle
(px = -pi sin(#) and px = pi cos(#)) at the entrance of the circuit. The current I in
Amps is given by
I0
1- h
, eI| 2 _
N , kOmeo-o #320
1
28(s - 1)!
2(s-1)
2
(2.39)
p
L, is the Form Factor of a waveguide mode, describing its interaction with an electron
beam, and N, is the Norm of a waveguide mode, accounting for the power in any given
transverse cross section.
Ls=
Ns =
[k(
kiL
47r
aX
+
a)
1i
|s|8
_y
(x, Y)
J{E, x H* - H, x E*}
S - zodSi
(2.40)
(2.41)
where X and Y are the transverse coordinates of the electron guiding center. It is
very useful to define a coupling factor
|L |
=
-Coupling
Factor,
(2.42)
which can be calculated in HFSS for different modes and beam radii to compare the
strength of the interaction. If we take the Equations (2.29)-(2.31), it is possible to
linearize them in the small signal regime and derive a dispersion relation that will
give the growth rate for a mode:
p3 _
F2
- Ir(n - b)F + (ft - Ab)I; = 0.
(2.43)
The three roots of F correspond to the growth rates of the three waves that will couple
with the electron beam. The three waves will propagate as e Fkz.
2.7
Short Pulse Propagation and Amplification
The pulse in the interaction circuit is expected to suffer from two dispersive effects:
waveguide dispersion from operating near the cutoff frequency of the interaction circuit and gain dispersion from the bandwidth-limited nature of the amplifier. Waveguide dispersion occurs because of the frequency dependence of the group velocity. It
has a significant effect because the frequency of the pulse is very close to the cutoff
of the waveguide (k, < k). A waveguide filled with an active medium (the electron
beam in this case) shows an additional dispersive effect due to that medium, especially near cyclotron resonance. This dispersive effect, referred to as gain dispersion,
is observed for pulse lengths with a spectral width on the order of or greater than the
bandwidth of the Gyro-amplifier.
In order to accurately describe the propagation of a pulse in a gyro-TWT we will
need to derive a non-stationary equation for the excitation of the waveguide. This
derivation is similar to the work of Kovlev et al. [32] which derives the non-stationary
equations for the excitation of waveguides. Let us define the excitation current jR
and the perturbed electric field ER and magnetic field HR:
- j(x, y, z,te-i
jR
ER
(2.44)
E(x, y, z, t)e-iwt
(2.45)
HR =H(x, y, z, t)e-wt
(2.46)
=
Maxwell's equations for this system take the form
V x HR=
V
DR
at
-i-j
(2.47)
xER
-(2.48)
where we define the electric and magnetic flux as
DR= J6()ER(z,w)e-dw
(2.49)
Jp(Hc,)HR(z, ()e-ictdcJ.
(2.50)
BR =
This allows us to have a permittivity and permeability that are a function of frequency.
We consider only one forward and backward propagating waveguide mode for this
system, given that the gyro-TWT is highly lossy for other modes and the electron
beam parameters are tuned such that it does not interact strongly with other modes.
The two modes will be identified as
E = A+Ei
(2.51)
H = A+Hi
(2.52)
where A is the instantaneous amplitude of the pulse envelope. The permittivity 6P)
and permeability p(Q) can be expanded as a series about a frequency w, in our case
the frequency of the pulse,
d~e
001
(2.53)
W)"
-
c(2)
n!
(2.54)
-
n)". (:
[p(W) =
n=O
Using the above definitions Equations (2.47) and (2.48) can be re-written as
00
(d
in
V x E = iwIpH +
n
i
V x H = -iweE -i
n=1
n
/p
(2.55)
dwr +n dn-)
T
1
t
do
+
ndn-1)C
u
t
+n j.
(2.56)
Using vector identities [37] defined as
V x AiHi
= Ai(V x H+) - Hi x (VAi)
=
V x A+Ei
-AjiwEE± +
Az
(2.57)
(2 x H+)
(2.58)
= A±(V x Ei) - E± x (VA±)
(2.59)
+
09Z
(2 x E±)
(2.60)
we simplify further to
Bz (2 x E±)
aA
az(2
dW
d"-p ) anH
±22dn-)
n=1
/dne
(o o"
x H±) = -i
n=1
+E
d"-1)
+ndo-71)
(2.61)
8tn
atn
.
(2.62)
Multiply Equation (2.61) by H_ and Equation (2.62) by E_ and add them together:
9A
(E_ x H+ - H_ x E+)2+
a(E_
x H_ - H_ x E_)
Bz
00
8" A+H+
(dw"
n=1
dwn + n dw"-1
n=1
atA
+ n dwn-1 )H
nd
H
-
t"
n-1E
n1
8tE
dn n
+
n d
-1)
E_
+ jE_
(2.63)
n
Integrating over the surface of the waveguide [32, 37] we define the Norm factor for
the waveguide to be
J {Hv x Es - Hs
where
oSv
is 1 if s = v and is 1 if s
/
x Ev} - EdS 1 = N,56,
(2.64)
v. We will consider the expansion Equation
(2.63) n=1,2. And we note that
I
(-pHH- + EEE_)dSi
= 0.
(2.65)
We simplify Equation (2.63) as
NaA+
Sz
I (-ptHH+
i d
+2dwJ
+cEE+)dS 1 A+
at
(-pHH++ EE_E+)dS
at 2 +
I jE-dS
1
(2.66)
If we use the following identities [32]
d pHdH+ = HH+--P
dw
do
(2.67)
d
dk
Ns d
Ha
dk
(2.68)
Ha(tjz)-
(2.69)
(cEE+ - ptHH+)dSi
=-
dw
=
then Equation (2.66) becomes
N
__A
+
-N
8 z
A+
az
+
dkz8A+
dw at
dkz aA+
dw at
+-
i d2 kz& 2 A+
+
Ns
2 dw 2 &t2
i d2kz a2 A+
2 dw2 at2
-
1
N8 J
.
jE_dS11
J -.E
dS1
(.0
(2.70)
(2.71)
Here we have separated the mode's behavior into two parts; on the LHS are
terms associated with propagation and the RHs is the interaction with the electron
beam. This interaction is extremely complicated for the specific electron distribution
in the case of gyro-TWTs. However, in [16] Chu shows that, for a continuous wave
gyro-TWT, with small signal theory the beam's interaction with the field is linearly
dependent on the axial field amplitude:
+
Ns
j -E*dS1 = G(w)A+.
(2.72)
In the linear regime the electron beam's interaction with the wave is only a function of
the frequency of the wave. G(w) could be determined theoretically from small signal
theory as described in the previous section or it could be measured experimentally
for a given operating point. The assumption of linear gain places some specific constraints on the system. The electrons must travel with an axial velocity (vi) that is
approximately equal to the group velocity (v,) of the pulse. This requirement ensures
that the bunched electrons will not become displaced from the pulse as it propagates
down the circuit. This assumption falls apart if the pulse is severely broadened. Additionally, the gyro-TWT must not be operating in a saturated regime for Equation
(2.72) to be true. If the group velocity and axial velocity of the electrons are not
similar or the amplifier is operating in a saturated regime the electron's location and
interaction with the pulse must be calculated instantaneously.
For a gyro-TWT operating in a linear regime a pulse's propagation through the
interaction circuit can be described by
dkz A+
+
dw at
i d2 k a 2 A+_
-
2 dw2 at2
-
(2.73)
G(w)A+
Similar to optical systems [1, 25, 24], it is convenient to deal with the dispersion in
the frequency domain, because of the pulse's inherent bandwidth and the frequency
dependence of G(w). Using the Fourier transform relationship:
[i(w
-
c0 )]A+(z, w - oo) = F.T. I
A+(z, t)]
(2.74)
we transform to the frequency domain
09
w - cj0,)
-A+(z,
az
[dkz
- CO) + 2 ddo2
2
+G(w)A+(z, w - W0 )
do
o)
2
A+(z, w -
0
)
(2.75)
A+(z, w - w0 ) is the frequency spectrum of the pulse envelope. If we are interested
not only in the envelope but in the instantaneous field, we can say:
a(z, w) = A+(z,
-a(z,
az
w)
dkz
=
-i
[kz(Oo)
dw
w - j,)e-ikzz
- c00) + -
2
2
2
(2.76)
0 )2 ]
a + G(w)a
(2.77)
The three terms with kz are the first three terms for the expansion of kz(w), giving
us our final result:
w) = -ikz(w)a + G(w)a
-yza(z,
(2.78)
To derive a simple algebraic expression describing the broadening of Gaussian pulses
we will revert to a temporal solution. First, we begin by expanding kz and G
09 a(z, w) = -i [kz(o)
dkz
+ 1 d kz2
2 dw
2
+do
W-0)2
a
1d2 G2
dG
CuO) + 2 dw
+ G(. 0) + dow
aa002
(2.79)
-
redefine the frequency spectrum of the pulse envelope
a(z, w) = A'(z, w -
(2.80)
cj)e-ikzz+Gz
and invert our transform from Equation (2.74)
,
az A/=
dkz OA'
dw Ot
i d2 kz 92 A'
2 dw2 at2
.dG A'
dw at
-
1d 2G a2A'
2 dw 2 at2
(2.81)
We can transform this equation to the moving frame of the pulse
(dkz
dw
aA
A'
i d2k
=(2 dw)2-
dG'\
(2.82)
dw/
1 d2G
2~d
2A'
(2.83)
t2
8W2
We define a broadening term 3"
d2k
1" = dw2
-A'
i
-13"
2
d2G
(2.84)
'dw2
/ a 2 A'
(2.85)
at2
The algebraic solution to this equation is known [24] for a Gaussian pulse. The output
width can be approximated as
r2 t =
i~n +|1"|212
where 1 is the length of propagation. We note that
(2.86)
$d
= vg. If we define
T
as the full
width half max of the pulse we can simplify & ~ A, where A is the difference of the
operand measured at full width half max.
'ti
2.8
2
(
'rn +
A-
1
+ A dG
2
(2.87)
Conclusion
A general explanation for the operation of electron cyclotron masers and gyro-TWTs
has been presented.
The bunching mechanism for both fast and slow waves in a
streaming electron plasma was presented. The nonlinear theory of operation was derived, discussed and analyzed. Numerical simulations were performed that provided
excellent agreement with published values. A quantum mechanical description of the
energy levels of the electron was discussed. This approach shows that the medium
is purely absorptive if it is fully nonrelativisitic. However, once the system is mildly
relativistic the system will undergo stimulated emission if the frequency of the electromagnetic field is slightly greater than the gyration frequency of the electron. The
dispersion relation for linear amplification in gyro-TWT was presented. The model
for the amplification of short pulses in the linear regime of a gyro-TWT was derived
from fundamental principles.
Chapter3
250 GHz Gyro-TWT Design
3.1
Introduction
Studies have shown how the promise of high gain and high output power in gyroTWTs is limited by the strong forward and backward wave gyrotron oscillations
near the waveguide cut-off [4]. A scheme of distributed loading was demonstrated
to suppress the gyrotron backward wave oscillations (gyro-BWO) and forward wave
gyrotron oscillations near the waveguide cut-off using fundamental mode interaction
circuits [13]. Operation in fundamental modes is not feasible at high frequencies, such
as 250 GHz, where the radius of the interaction structure would be on the order of
a fraction of a millimeter. This results in high ohmic losses that can lead to thermal
damage and reduced gain. It also presents a significant challenge in transporting
an electron beam over long distances (~30 cm) without causing beam interception
on the waveguide walls. Operation in higher order modes poses its own problems
with oscillations at lower order modes. To prevent these oscillations a mode selective
interaction circuit with distributed loss that is easy to manufacture is highly desirable.
The danger of oscillations from lower order modes is easily seen in Figure 3-1 where
the dispersion relation for several modes in a cylindrical waveguide is plotted. One
candidate mode for an amplifier at 250GHz would be the TE0 3 . However, when
the electron beam line for a grazing condition is considered it becomes apparent that
oscillations are possible where the beam intercepts with other modes. A forward wave
oscillation with the TE 23 mode or a backward wave oscillation with the TE81 and TE4 2
modes are the most significant. Clever selection of electron beam parameters, the
inclusion of severs or distributed loading can increase the start current for oscillations,
but any realistic design must have an interaction circuit that directly addresses the
presence of lower order modes.
270
260
250
240
230
220'
-3000
-2000
-1000
0
kz
1000
2000
3000
Figure 3-1: Dispersion relation for a circular waveguide with a 2 mm radius.
The design of a 250 GHz gyrotron traveling wave amplifier for nanosecond pulses
with gain exceeding 50 dB that focuses on the presence of lower order modes is
presented in this chapter. The amplifier uses a novel photonic band gap (PBG) interaction circuit with mode selective properties. The PBG interaction structure confines
a TE03 -like mode with comparable interaction efficiency to a circular waveguide, while
providing lower-order mode suppression and distributed loss along the entire interaction circuit. The design of the input coupler, output coupler and electron gun will
also be discussed in this chapter.
3.2
Attenuation in Waveguides
In order for the gyro-amplifier to operate correctly self start oscillations, driven oscillations and BWOs must all be subdued. The traditional method for suppressing
these oscillations consists of minimizing reflections in the amplifier circuit to lower the
diffractive
Q and
placing severs (short sections of highly attenuating waveguide) to
increase the start current for these oscillations. Both of these features are complicated
to design and manufacture for very high frequency systems. Severs can also waste
valuable space in the flat top of the magnetic field thereby decreasing the performance
of the amplifier. An alternate approach is to design an interaction circuit that has
an attenuation constant that is highly frequency dependent in such a way to prevent oscillations at lower frequency modes and near cutoff for the operational mode.
Traditional waveguide circuits do have a frequency dependence in the attenuation
constant especially near cutoff, but it is insufficient to suppress these oscillations. In
a circular waveguide the electric and magnetic fields for TEn, modes can be written
as:
=
E
E
kp
ke
A cos(n#)Jn(kep)e-ik,
A sin(n#)J'(kep)e-ikz,z
Ez = 0,
Hp=
HO =
Asin(n#)J'(kcp)e-ikz,
ikzn A cos(n#)Jn(kcp)e-ikzz
k2p
Hz = A sin(n#)Jn(kep)e-ikzI
where kc = unm/a, vnm is the mth root of the nth order Bessel function and a is
the waveguide radius. In order to calculate the attenuation constant (oc) we must
find the power propagating down the waveguide (PO) and the power dissipated on the
conducting walls (P) of the waveguide. The power propagating down the waveguide
is found by integrating the Poynting vector over the cross sectional surface of the
waveguide
PO =Re
2
1a
27r
p=0 =0
t - pdodp
R x fA*
(3.1)
where i is the direction of propagation. The dissipated power is found by integrating
the surface current
J,
on the surface of the waveguide at that cross section
-j
P.iP-Rs
jd
|J2ad#$
27rjx
(3.2)
where R, is the surface resistivity and |J |2 is given by the magnetic fields on the
surface [lHg|2 + H2| 2 ]. Once P and P are known the attenuation constant can be
defined as
a
=P
(3.3)
S2P
Due to the surface currents and the propagation constant kz the attenuation constant is frequency dependent and mode dependent. For waveguides with complicated
geometries like confocal or photonic band gap structures it is not possible to analytically calculate the attenuation constant because the fields of the mode are not well
defined. One must use a numerical code, such as High Frequency Structure Simulator (HFSS), to observe how power propagates through the waveguide as a function of
frequency. An ideal interaction circuit would confine the operating mode with limited
loss in the amplification region and have an attenuation constant that is very high
for modes that are not of interest. Mode selective circuits can be made in a variety
of ways (dielectric loading, confocal, PBG, etc.) that often have a detrimental impact on the mode of interest as well as additional complexity in fabrication. Finding
a balance is the key to developing a successful overmoded interaction circuit for a
gyro-TWT.
3.3
Photonic Band Gaps
It is possible to create a large variety of photonic band gap structures in one, two
and three dimensions that support TE, TM and TEM waves. A photonic band gap
of interest for gyro-TWTs needs to support TE modes that will not diffract in the
transverse direction with respect to the electron beam. Furthermore, if the circuit is
to be overmoded then the lowest band gap for a given set of lattice constants should
not extend to zero frequency. Taking these things into consideration we have selected
a two dimensional triangular metallic lattice (Figure 3-2) for the gyro-TWT and it
will be the only one discussed presently. We begin by defining the conductivity profile
of the lattice as
o- Tm
oTM,, = (mn + -
2
bx +
-nbey
2
(3.4)
(3.5)
where b is the lattice spacing, m and n are integers and xi = xed + yy.
We will define our fields as a function of sF1 , z, t. Due to the symmetry and
linearity of the system we can assume that the axial wave number kz and the frequency
w are fixed. We write the Helmholtz wave equation as
VI@ (i)
with the boundary condition
=
(k
-4
,(z)
8@
-- is = 0
(3.6)
(3.7)
where n' is the vector normal to the metallic surface S. The discrete array of metallic
rods has a solution that can be written in Bloch form
1/'
+(
=
@ (i) L ) ei
TTmn)
'"m
(3.8)
where ki = kx8 + ky e is the transverse wave number, allowing us to solve the wave
.......
....
:1-1-1111
.....
...........
n.,.....
0
--
7
X
Figure 3-2: Triangular lattice of metallic rods. The fundamental unit cell is marked
with a dashed line.
1
J
-W----I
b
Figure 3-3: Reciprocal triangular lattice where b' =
ducible Brillouin zone.
_.
The gray area is the irre-
equation in the fundamental cell in Figure 3-2 given by
X -- Y
(3.9)
<b
<3_- ,y11
y <-b43 b.
The discrete nature of the system also imposes periodic boundary conditions:
-+ I y y
2 V/3
x, V5b )
= eikxb4V
ikxb/2+ikyv5b/2
2
x
+
-
V3
,
y, y
-
(3.10)
(3.11)
vK-).
Equation (3.8) imposes limits on the value of kI, shown as the Brillouin zone in
Figure 3-3. The three special points F, X and J are at k1 = 0, kic = (27//5b) y and
k1 = (27r/x/5b)(82 + 1/v/5y).
One possible method to solve the wave equation in the unit cell area is using
HFSS [41]. By imposing the proper boundary conditions we can pre-select for TE
modes and fully explore the possible k-space of the system. For any given k 1 there
will be a corresponding phase advance across the unit cell. Perfect H boundaries are
defined on the top and bottom of the unit cell to limit eigenmode solutions to TE
modes. The wave number can be specified by creating a master/slave boundary as
shown in Figure 3-4. The master/slave boundary defines the phase advance across
the two orthogonal directions of the unit cell thereby specifying the wave number.
For example, going from the F point to the X point, on Figure 3-3, k. will always
be zero, but 0 < ky < 2ir/v/5b. If we consider the periodic boundary conditions
for this range of values, Equation (3.10) indicates that the phase shift
q1
= 0 from
the master #1 to the slave #1 boundary and Equation (3.11) indicates there is a
0
#2
< 0
i phase shift from the master #2 to the slave #2 boundaries in Figure 3-4.
Performing the same analysis from X to J and J to F will complete the exploration of
the Brillouin zone. The necessary values of k, and ky and the corresponding phases
of the master/slave boundaries are listed in Table 3.1.
slave #2
master #2
(b)
(a)
Figure 3-4: (a)Master/slave boundaries for the unit cell and (b) the equivalent HFSS
model with the complex magnitude of the electric field of the lowest order propagating
mode for a/b = 0.2, #1 = 7r/9 and #1 = 191r/18.
Table 3.1: Simulation Parameters to Investigate Brillouin Zone
F to X
X to J
J to F
kx
ky
0 -+ 27r/N/3b
0
27r/v/b
0 -+ 27r/3b
27r/3b -+ 0 27r/V b -+ 0
01
0
0 -+ 27r/3
27r/3 -> 0
$2
0 -> 7r
-+47r/3
47r/3 -> 0
7r
The results of eigenmode simulations for a/b = 0.2 and the full range of values
in Table 3.1 are shown in Figure 3-5. The 8 lowest order modes are plotted as a
function of ki and normalized frequency. A photonic band gap is defined as a range
of frequencies over which no mode has a solution. In Figure 3-5 we observe that a
very small gap opens from wb/c = 6.69 to wb/c = 6.75. Figure 3-5 is in excellent
agreement with results published by Smirnova et al. [45] using alternate methods. In
Figure 3-6 the eigenmode frequencies are plotted for a/b = 0.43 which corresponds to
the radius to spacing ratio of the design mode. In this figure we can see three photonic
band gaps for wb/c = 5.35 -> 6.62, wb/c = 11.37 -- 12.12 and wb/c = 15.26 -> 16.93.
Performing HFSS simulations over the full range of a/b it is possible to produce
the global photonic band gap plot for a triangular lattice shown in Figure 3-7. This
12
- -
---.-.-
10
- --
8 -... .- ..
4-2
0
x
IF
J
Figure 3-5: Normalized eigenmode frequencies as a function of k1 for a/b = 0.2.
18
.
16
1 4..
..
.-
.
.... . .
. ....
12-
10
-10-------
------ ----------
8 8 . .. . . . . .
Figure 3-6: Normalized eigenmode frequencies as a function of ki for a/b = 0.43.
61
band gap plot is also in excellent agreement with results published by Smirnova et
al. [45].
18- ........................
..........................
15- ........................
......
.....
.........
..
.
................
12-
...........
3|F ...............
0.1
. .. ... .. .. .. .. ..
0.2
0.3
0.4
0.5
a/b
Figure 3-7: The global band gap plot for a triangular lattice, where a is the spacing
and b is the diameter of the rod.
3.4
PBG Interaction Circuit
The interaction circuit is the most sensitive component in the entire design process
and in large part defines the standards that other elements of the gyrotron amplifier
must meet. The interaction region is limited to the flat top length of the magnetic field
defined by 0.1% variation in the magnetic field (30 cm for this experiment). In order
for the amplifier to interact efficiently with the electron beam, the beam must undergo
a phase bunching process that results from the interaction with an electromagnetic
field. After the phase bunching process is complete a majority of the power that will be
extracted from the beam is produced in a few cenitmeters. It is important to quickly
terminate the interaction with the beam to prevent reabsorption of microwave power.
The selected photonic band gap (PBG) waveguide is a novel interaction structure for
62
gyro-TWTs that is highly mode selective and has strong coupling with the electron
beam resulting in high interaction efficiency. A variety of geometries have been studied
to find suitable mode confinement and frequency dependent characteristics.
The
circuit is based on a PBG composed of a two dimensional triangular lattice of metal
rods with a central defect and it will be shown that it can meet the requirements
of mode selectivivity and high interaction efficiency.
The advantage of using this
PBG structure is that its band gap structure is favorable for supression of lower
order modes. With the proper lattice constants the PBG can confine electromagnetic
radiation at 250 GHz while suppressing both higher and lower frequencies.
The
dimensions of the lattice are tuned so that the lattice acts as a perfect reflector in a
narrow band of frequencies around the operating mode. A defect is created in the
lattice by removing some rods, allowing a higher order mode to be confined with a high
quality factor (Q). Other modes that can exist in the defect either at higher or lower
frequencies suffer significant losses because of the partially transparent lattice [45]. A
PBG gyrotron oscillator has already been demonstrated confirming the advantages
of such a circuit in overmoded operation [43].
A transverse cross section of a triangular lattice of metallic rods is shown in Figure 3-2. The properties of the lattice are governed by the radius a and spacing b of the
rods. The fundamental unit cell is demarcated by a dashed line. The fundamental
cell should be infinitely repeated to create a perfect PBG. However, a non-infinite lattice still exhibits the basic nature of the PBG with highly reflective bands, especially
in the case of metallic lattice. The corresponding band gap plot for the triangular
lattice in Figure 3-2 is shown in Figure 3-8. The enclosed regions are the band gaps of
the lattice. An electromagnetic wave of a given frequency will be perfectly reflected
by a lattice with corresponding lattice constants a and b that reside in a band gap.
If the frequency does not reside in the band gap the wave will be partially or fully
transmitted through the lattice. The design point for the experiment at 250 GHz
is marked with a red dot. The lower edge of the gap was selected because we are
concerned about oscillations from lower order modes and this guarantees that none
12
9
3
0
0
0.1
0.2
0.3
a/b
0.4
0.5
Figure 3-8: The global band gap plot with the operational spacing of a/b = 0.43 is
marked with a dashed red line. The TE0 3 mode for a finite lattice with three rods
removed is marked with a red point.
will be confined.
One difficulty of using a PBG structure is finding a mode that will interact well
with an electron beam. By removing a number of rods a defect region is created
in the lattice whereby a mode can be confined. Starting with the central location
and removing three concentric rows of rods a TE 03 -like mode can be excited. This
mode is similar to the TE03 mode in a lossy circular waveguide and will have a good
interaction with an electron beam that has a 1.1 mm radius that would reside on the
second radial maximum of the electric field. This structure was simulated using the
eigenmode solver in HFSS to find the confined modes. In Figure 3-9 the TE0 3 mode
is plotted in a circular waveguide with a radius of 2 mm. This waveguide has many
lower order modes that could easily spark oscillations in particular the TE 2 3 , TE8 1
and TE 4 2 . The corresponding TE0 3 -like mode confined be a PBG waveguide is shown
in Figure 3-10 with a cutoff at 242.68 GHz, the PBG lattice constants of wb/c = 4.87
and a/b = 0.43 provide the appropriate amount of confinement and suppression of
other modes. The diameter of the defect region is approximately the same size as
the circular waveguide.
Unlike the TE03 mode in a circular waveguide, the main
mechanism for attenuation is not ohmic losses, but diffractive radiation through the
lattice due to the finite nature of the lattice. This should minimize the amount of
ohmic heating on the metallic rods and it will require the addition of lossy material
after the lattice to absorb the stray radiation.
Using a combination of eigenmode and driven modal simulations in HFSS it is
possible to calculate the attenuation in the waveguide as a function of frequency as
shown in Figure 3-11. With a loss of approximately -2dB/cm at 250 GHz we have a
reasonable amount of attenuation in the waveguide. Additionally, HFSS simulations
indicate the TE 23 , TE81 and TE 42 modes are not supported in the PBG waveguide.
HFSS can also be used to calculate the coupling between a waveguide mode and the
electron beam. The normalized coupling factor [37] from Equation (2.42) for the TE0 3
mode in a circular and PBG waveguide are plotted relative to each other in Figure 312. The region of interest is the second maximum in the coupling factor, because an
electron beam radius of less than 1 mm is impractical for this experiment. For an
electron beam radius of 1.1 mm the normalized coupling factor for the PBG waveguide
is 48% compared to a circular waveguide. The decrease in the normalized coupling
factor between the electron beam and the PBG waveguide mode is acceptable in light
of the additional advantages of the PBG waveguide. Furthermore, the interaction
between the electron beam and a waveguide mode can be increased by adjusting
electron beam parameters. The normalized coupling factor of the PBG waveguide is
almost constant between 1-1.2 mm allowing for some variation in the electron beam
radius without impacting the coupling.
4mm
Figure 3-9: The complex magnitude of the electric field for the TE03 mode confined
in a circular waveguide.
10 mm
Figure 3-10: The complex magnitude of the electric field for the TE03-like mode
confined in a PBG composed of a triangular lattice of metal rods.
..
..........
...........................
. ...........................
...............
....
E
I-%
O2
'
245 247.5 250 252.5 255 257.5 260
Frequency (GHz)
-6
Figure 3-11: Attenuation as a function of frequency for the TE03 -like mode.
normalized L-facor
1
*. -_-PBG
-; 0.8-
-Cylindrical
Wav
0 0.70
0)0.6o- 0~0 0.4..N 0.3
E 0.2
Z0.1
0
0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Beam Radius (mm)
2
Figure 3-12: Normalized coupling coefficient comparison for a circular waveguide and
the PBG waveguide. The electron beam radius for the gyro-TWT will be 1.1 mm.
3.5
Linear Growth Rate
In the limit of small signal gain the equations of motion for the electrons in the
gyro-TWT, Equations (2.29)-(2.31), can be linearized [38] to produce the dispersion
relation in Equation (2.43). Using this small signal dispersion relation in combination with the normalized coupling coefficient, loss parameters for the PBG circuit,
and S-parameters for circular to PBG waveguide junctions calculated in HFSS it is
possible to calculate the single pass and round trip gain for the device. The linear
growth rate is extremely dependent on the electron beam parameters and needs to be
optimized to maximize the circuit gain and bandwidth. It is easiest to achieve large
bandwidths when the amplifier is slightly detuned from a grazing condition with the
waveguide dispersion relation. The further the grazing condition is from cutoff the
larger the bandwidth, but lower the gain. There are also several disadvantages that
come with operating too close to cutoff. First, a strong interaction between the wave
and the beam close to cutoff increases the possibility of self start oscillations. Second,
waveguide dispersion limits the gyro-TWTs ability to amplify short pulses, as will
be discussed in detail in Chapter 4. For these reasons the amplifier's bandwidth is
centered at 3% above cutoff or 250 GHz. There are also some advantages to lowering
the linear growth rate (for a fixed attenuation constant) in order to make a high gain
amplifier while avoiding driven oscillations. If the growth rate is relatively small a
long circuit will be required to achieve the desired gain of more than 50 dB. For a
driven oscillation to occur, the roundtrip gain must be greater than 0 dB, meaning
a wave that is amplified, reflected from the end of the interaction region, propagates
to the start of the circuit and is reflected again has increased in energy. If the gain
per cm is approximately twice the attenuation per cm it is easier to suppress the
driven oscillation. With an attenuation of -2dB/cm at 250GHz a linear growth rate
of 4dB/cm meets this condition.
Taking these things into consideration the opti-
mized electron beam parameters that were calculated for this structure are listed in
Table 3.2.
Figure 3-13 shows the linear growth rate as a function of frequency for the TE03
Table 3.2: Design Operating Parameters from Linear Theory
Beam voltage, Vo
Beam current, Io
Beam pitch factor, a
Beam Radius
Input power
Center Frequency
Peak Magnetic Field
Operating Mode
Circuit Length
32 kV
1.0 A
0.75
1.1 mm
1 mW
250 GHz
8.81 T
PBG TE0 3 -like
22 cm
mode in a circular and PBG waveguide for the same electron beam parameters, calculated using Equation (2.43) and the parameters in Table 3.2. These operational
parameters provide a reasonable growth rate, a very large bandwidth and no amplification within 2 GHz of the cutoff frequency. As was mentioned previously the
normalized coupling factor is considerably lower for the PBG waveguide compared to
the circular waveguide, but it has a limited impact on the growth rate.
It is reasonable to assume that the amplifier will be operating at the edge of
the linear regime so we can use the linear growth rate for an estimate of the circuit
gain. For a fixed input power of 1mW, of which only 1/3 couples into the growing
solution of the dispersion relation in Equation (2.43), the circuit gain is calculated
by subtracting the attenuation from the growth rate and multiplying by the circuit
length, with the results show in Figure 3-14. In addition to the single pass gain, it is
important to calculate the round trip gain to consider the effect of driven oscillations.
As shown in Figure 3-14 the round trip gain is below 0 dB for the full operational
range of the amplifier.
In order to include the effects of perpendicular velocity spread of the electron beam
MAGY [101, a non-linear, self-consistent simulation code, was used. This software is
the most common package used for the design of gyrotron oscillators and amplifiers,
and has shown good experimental agreement for both cases [10] [31]. In Figure 3-15
......................................
. .....
...
. . ........
65S4.
3
2
1
0
245
242.5
252.5
250
247.5
Frequency (GHz)
255
257.5
260
Figure 3-13: Gain in dB/cm from linear theory.
C
3020100
-10-
242.5
245
247.5
255
252.5
250
Frequency (GHz)
257.5
260
262.5
Figure 3-14: Gain in the device calculated from the linear growth rate and loss per
unit length of the circuit. Reflections are suppressed enough to prevent oscillations.
the gain in dB/cm is shown for the linear growth rate and simulations using MAGY
at 250 GHz. A practical electron gun should be able to achieve 5% perpendicular
velocity spread as will be shown in Section 3.8. The electron beam in the MAGY
simulations presented includes 5% velocity spread and is otherwise identical to the
parameters listed in Table 3.2. In addition to the previous calculations using the
linear growth rate and a circuit length of 22 cm, Figure 3-16 shows the results of
MAGY for a 26cm cavity. The frequency dependent characteristics of the structure
that are calculated using HFSS were implemented in MAGY. MAGY simulations
require the use of a circular waveguide and the geometry cannot be altered to take
into account the PBG. To account for this the electron beam was placed at a slightly
larger radius where it would have the same coupling coefficient as the PBG mode.
To achieve the correct attenuation constant the conductivity of the waveguide in the
interaction region was altered to match the the values in Figure 3-11. The circuit in
MAGY also includes an uptaper designed using CASCADE to minimize reflections,
discussed in Section 3.7. The input power for the numerical simulations was 1 mW.
Agreement between linear theory and MAGY is excellent, with both cases showing
approximately 3 GHz of bandwidth. The center frequency of the MAGY simulations
is slightly lower than the linear theory due to the impact of velocity spread which is
more severe on higher frequencies.
The output power of the amplifier for a 1 mW input and for saturated operation
is shown in Figure 3-18. For an input of 1 mW the amplifier produces more than 1
kW of peak output power. The saturated output power shows that the amplifier is
operating on the edge of the saturated regime. Figure 3-19 shows the input versus
output power curve for 250 GHz. Below 1 mW of input power the amplifier exhibits
linear behavior and above 1 mW the gain falls rapidly as the output of the amplifier
saturates.
---
-
-
-
I
--
-
-
--
---
- --
--
2.5F
9
2.4-
*
2.3
.. 0..
.
2.2
2.1
Linear Growth Rate
MAGY
S
0
1
2
3
4
v1 Spread (%)
5
6
7
Figure 3-15: Gain in dB/cm at 250 GHz as a function of perpendicular velocity spread
calculated from the linear growth rate and MAGY.
Table 3.3: Design Operating Parameters from MAGY
Beam voltage, Vo
Beam current, 1
Beam pitch factor, a
Beam Radius
Input power
Center Frequency
Peak Magnetic Field
Operating Mode
Perpendicular Velocity Spread
Circuit Length
32 kV
1.0 A
0.75
1.1 mm
1 mW
250 GHz
8.81 T
PBG TE0 3 -like
5%
26 cm
" so
-
- - ,
..
......
.
504540
247.5
250
252.5
Frequency (GHz)
255
257.5
Figure 3-16: Gain in the device calculated from the linear growth rate for a 22 cm
circuit compared to results from MAGY for a 26 cm circuit with 5% perpendicular
velocity spread.
.............
..................
................
20
Distance (mm)
30
Figure 3-17: Dimensions of circuit simulated in MAGY.
40
1200
1000
800
0
600
400-
200-
/
--
in1 mW
+ Saturated Output Power
947 248 249 250 251 252 253 254 255 256 257
Frequency (GHz)
Figure 3-18: Output power as a function of frequency simulated with MAGY using
a 32 kV, 1 A electron beam with 5%vi spread.
103
102
C.
0
101
10~-6
10-5
10- 4
10-3
10-2
P. (W)
Figure 3-19: Output power as a function of the input power for 250 GHz.
3.6
Input Coupler
The input coupler must efficiently convert the 250 GHz driver signal from TE 10 mode
in a fundamental rectangular waveguide into the operational mode of the amplifier.
The operational mode of the amplifier is very similar to the TE03 mode in a circular
waveguide. The proposed method for the efficient excitation of the amplifier is the
use of a wraparound coupler for a small section of circular waveguide, with a radius
that corresponds to the same cutoff frequency of the PBG waveguide, that mates
directly to the PBG structure.
A wraparound coupler works by exciting a standing wave in a rectangular resonant
ring that wraps around the circular waveguide. Coupling slots are placed at nulls in
the electric field of the ring resonator to induce currents in a circular waveguide that
will excite the TE0 3 mode. Once the mode is excited in the circular waveguide as
much energy as possible should be directed towards the interaction region. To direct
the mode towards the PBG structure a down taper is placed on one side of the
coupling slots. The down taper reduces the size of the circular waveguide until it
goes below cutoff for the TE0 3 mode. This reflects the power traveling towards the
electron gun without producing mode conversion. The additional benefit of having
a down taper is the electron gun is protected from high power radiation in the TE0 3
mode that could be generated if the amplifier begins to oscillate. Simulations of the
structure were run in HFSS using driven modal solutions with an input waveport on
the rectangular waveguide leading to the wraparound coupler. An output waveport
was placed at the end of 5 mm of PBG structure and short cylindrical section. This
allows the S-parameters to account only for the pure TE03 content of the waveguide.
An electric field symmetry plane was used to reduce simulation time. The geometry
is symmetric about the x-z plane.The schematic of the wraparound coupler is shown
in Figure 3-20(a) as it was modeled in HFSS. In Figure 3-20(b)-(d) the evolution
of the complex magnitude of the electric field is shown as it propagates down the
coupler. The wave begins by forming a standing wave in the resonant ring that then
....
..
...............
b)
a)
C)
d)
Figure 3-20: The (a) cross section the input coupler with the components (from
left to right) downtaper, wraparound, circular waveguide and PGB waveguide. The
complex magnitude of the electric field at 250 GHz shown in (b) the wraparound, (c)
the circular waveguide and (d) the PBG waveguide.
%A
E
UQ.
a)
-21
247
248
249
250
Frequency (GHz)
251
252
253
Figure 3-21: S-parameters for the wraparound input coupler.
couples into the circular waveguide where the TE03 mode is formed. After the TE0 3
propagates into the PBG waveguide it changes into a form that is very similar to
the result in Figure 3-10. The S-parameters of the coupler are shown in Figure 3-21.
Using HFSS we calculated the S1 for the input coupler to be -18 dB at 251 GHz.
S21 is measured after the wave has propagated into the PBG waveguide. The coupler
converts more than 50% of the power into a forward propagating wave over 2 GHz of
bandwidth.
3.7
Output Coupler
Power extraction from the amplifier must meet two criteria: minimum reflections into
the interaction circuit and a radiation pattern that will couple into a transmission
line.
As a first stage of the output coupler the near cutoff interaction circuit is
enlarged to lower the cutoff frequency, decrease the interaction with the electron beam
and minimize reflections. Using CASCADE a multi-step uptaper was optimized to
minimize reflections from cutoff up to 260 GHz. The final radius of the waveguide,
a = 3.88 mm was selected so that the bounce angle would be OB
50' determined
by
0B
=
sirF 1 (vnm/ka)
The uptaper simulated in CASCADE is shown in Figure 3-22.
(3.12)
Both ends of the
waveguide are perfectly matched in this simulation and do not produce reflections.
S11 and S 21 for the TE0 3 mode in a circular waveguide are shown in Figure 3-23 and
Figure 3-24, respectively. After the uptaper, the TE0 3 mode can either be extracted
axially or it can be mode converted into a Gaussian beam and be extracted radially.
The axial configuration is simplest, but not optimal for pulsed DNP experiments as
they require a Gaussian beam to propagate the radiation to the NMR test stand and
couple it into the sample [3]. The Gaussian beam mode converter can be designed
by considering the quasi-optical nature of the 250 GHz signal and decomposing the
........... ....... ..
. .... .... ... .. . . .. .... ...
....
.......
E1
E
w0
...
......
0
10
20
30
40
Distance (mm)
50
60
70
80
Figure 3-22: Uptaper dimensions simulated and optimized in CASCADE.
-5-10
-15-20-25-
245
247
249
253
251
Frequency (GHz)
255
257
259
Figure 3-23: S11 of the uptaper for the TE 03 mode.
243
245
247
249
251
253
Frequency (GHz)
255
257
Figure 3-24: S21 of the uptaper for the TE03 mode..
259
waveguide mode into a series of plane waves that propagate at a bounce angle based
on the radius of the waveguide. If the waveguide was terminated these plane waves
would radiate outward radially permitting a parabolic mirror to reflect them as a
plane wave. In order to limit the direction of the radiation the termination is not a
smooth cut, but a notch in the waveguide that allows all of the exiting plane waves
to radiate into a half-space.
3.8
Electron Gun
The electron gun must meet very specific requirements in order for the amplifier circuit to operate successfully, because the interaction efficiency is highly dependent
on the beam quality. In a standard electron gun a large electric field extracts electrons from an annular ring on the cathode. The extraction of electrons occurs by
heating the cathode and applying a large potential bias between the cathode and
anode. The electric field between the cathode and anode is partially transverse to the
static axial magnetic field at the cathode. The transverse component causes the electrons to begin gyrating. The electrons undergo compression as they enter the higher
intensity magnetic field area. The static magnetic field also prevents the electrons
from being intercepted by the anode which is before the interaction region. There
can be either one (diode) or two (triode) anodes in the electron gun. The triode
configuration increases the beam quality by creating extraction and acceleration regions. Additionally, the ability to modify the voltage on the first anode (mod-anode)
provides increased control over the static electric field profile allowing the user to
vary the alpha of the electron beam. This is of great importance in the operation
of the gyro-TWT as it allows the user to tune the operating region of the amplifier
by raising or lowering the main magnetic field and adjusting the alpha accordingly
to reach grazing condition. The electron gun was designed using the simulation code
MICHELLE. Initial design parameters for the electron gun configuration were calculated using equations based on conservation of angular momentum and adiabatic
Table 3.4: Electron gun parameters
Electron Gun
Type
Beam Voltage
Beam Current
Beam Radius
Velocity Pitch Factor, a
Perpendicular Velocity Spread
Cathode Radius
Compression Ratio
Magnetic Field at Gun
Electric Field at Gun
Cathode - Main Anode Voltage
Preliminary Parameters
Triode
35 k
2A
1.0 mm
0.75
< 3%
4.4 mm
19
0.46 T
< 100 kV/cm
15 kV
electron motion.
Using the design equations given in [2] a first order design was found by specifying
the main design parameters: current, voltage, main magnetic field, compression and
velocity pitch factor, oz. Experience has shown [34] that a slope of 50 degrees (between
the axis of symmetry and the emitter surface) for the emitter produces the highest
quality beams with an acceptable beam width. The basic gun design parameters can
be plotted as a function of the mean emitter radius as shown in Figure 3-25. In order
to minimize the maximum electric field and decrease the width of the emitter strip a
large compression factor is favorable. The practical limit for the compression factor
is approximately 25, setting this as the criteria we arrive at a mean emitter radius of
5.2 mm. The remaining design parameters are marked as black dots in Figure 3-25.
The initial drawing of the triode electron gun using the design parameters from
Figure 3-25 for a cathode slope of 50 degrees was optimized using MICHELLE to
minimize the perpendicular velocity spread.
MICHELLE also verified the perfor-
mance of the electron gun over a range of beam voltage(up to 50 kV) and mod-anode
voltage to ensure that the electron gun would have the desired tunability. The final
CAD drawing of the electron gun is shown in Figure 3-26. The mean emitter radius,
emitter strip width, cathode slope and cathode to anode spacing correspond to the
0.8
0.6
=
0.4
0.2
0
0.2
0.4
0.6
0.8
rc(cm)
(b)
0.6
rc (cm)
(d)
0
0.2
0.4
0.6
0.8
rc (cm)
(e)
Figure 3-25: Gun parameters calculated using equations derived by Baird and Lawson
for a 32 kV 2 A beam and 50 degree cathode tilt. The (a) compression, (b) magnetic
field at the emitter, (c) emitter width, (d) maximum electric field and cathode to modanode potential, (e) Langmuir space charge, and (f) cathode to mod-anode spacing,
d, and guiding center spread are plotted versus the mean emitter radius. Black dots
mark the selected design values.
Figure 3-26: Schematic of the electron gun design with all units in mm unless otherwise marked.
design parameters shown in Figure 3-25. The drawing of the electron gun includes the
cathode (blue line), emitter (teal line), mod-anode (red line), the anode or common
(green line) and dielectric spacers (purple line). The full CAD drawing as modeled
extends to the right until the beginning of the interaction circuit and the electron
gun is cylindrically symmetric about the black line. The electron gun placement
was determined using the magnetic field profile of the 9.6 T Cryomagnetics magnet
currently under construction. The emitter strip is located 366.8 mm before the beginning of the flat top of the magnetic field. One important consideration is that
the magnetic field is monotonically increasing from the emitter location to the flat
top. The magnetic field used to model the electron gun and the guiding center radius
is plotted in Figure 3-27. The guiding center radius is monotonically decreasing as
it approaches the flat top of the magnetic field. In Figure 3-27 the flat top of the
magnetic field extends from 0 to 30 cm. The location of the electron gun is marked
with a black dot. For the current design the magnetic field uses an internal gun coil
at 25% of its rated current to decrease the magnetic field at the cathode. It is possible
that during operation the gun location could be changed requiring an adjustment of
the current in the gun coil to maintain the compression and ensure a proper guiding
center radius. The magnetic field ±25% the rated current of the gun coil is shown
in Figure 3-28. The magnetic field at the cathode for the optimized electron gun is
marked with a black dot.
The physical location of the electron gun will reside comfortably within the 5 in
magnet bore. A side view of the 9.6 T Cryomagnetics magnet is show in Figure 3-31.
The emitter strip will be located 1 cm into the bore of the magnet. The electron gun
will be encased in a metallic can to provide a grounding plane allowing for the option
of moving the electron gun without affecting the grounding plane by the presence of
the inner wall of the magnet bore.
The optimal design parameters of the electron gun consist of a 2 A beam with a
cathode voltage of -32 kV, a mod-anode voltage of -18kV, and an anode voltage of
................................................................
....
E
N
420'
-40
-20
0
20
40
60
z (cm)
Figure 3-27: The magnetic field as used to model the electron gun performance. The
guiding center radius is also plotted along the axis of the magnet.
0.8
0.7-
1
1
Main Field
Main Field + 25% Trim
Main Field - 25% Trim
0.6-
N0.5-
0.40.3 0.2
-45
-40
-35
z (cm)
-30
-25
Figure 3-28: The main magnetic field with the gun coil magnet at 0% and +25% of
the rated current.
Electric Field (kV/cm)
65.1
48.8
32.6
16.3
0.00
Figure 3-29: Electric field distribution for the cathode at -32 kV, mod-anode at -18kV
and anode at ground.
Electric Potential (k)
0
-8
-16
-24
-32
Figure 3-30: Electric potential lines and particle trajectories for the cathode at -32
kV, mod-anode at -18kV and anode at ground. Gun coil at 25% of rated current.
0
V. The electric field produced under these conditions is shown in Figure 3-29 with
a comfortable maximum of 65.1 kV/cm. The potential lines and particle trajectories
under the same conditions are show in Figure 3-30.
Two distinct regions can be
observed in the electron gun. The first stage extracts the electrons with the field
produced by the mod-anode and is approximately one cyclotron orbit in length. The
a and velocity spread are primarily determined in this section.
The accelerating
region occurs after the first complete cyclotron orbit where the electrons pass over
potential lines that are transverse to their motion.
The triode electron gun designed using MICHELLE produces the desired beam
pitch factor of 0.75 with a velocity spread of less than 2% for a 32 kV 2 A beam
and 18 kV mod-andode voltage as shown in Figure 3-32.
The electron gun was
also investigated over a large range of mod-anode and beam voltages with results
in Figure 3-32, Figure 3-33 and Figure 3-34 indicating that this design is capable of
producing an a of 0.75 for beam voltages as high as 50 kV with a velocity spread
of -2%.
The velocity spread in the constructed electron gun will be greater due to
surface roughness effects and machining errors and could rise to as much as ~5%.
The current design consists of a cathode with a 50 degree tilt that can comfortably
produce a 2 A beam (current density of 5 A/m 2 ). The gun coil will provide additional
tunability of the beam pitch factor and aid in the suppression of velocity spread. From
32-50 kV varying the mod-anode voltage to be 50-66% of the cathode voltage varies
the beam from an a of 1-0.6 without ever exceeding the 96 kV/cm limit or reflecting
electrons.
............
Gun
365[14.38in]'-Z
SCALE 0.09: 1
2.
0252 [9.92in]
Figure 3-31: Cross sectional CAD drawing of 9.6 Tmagnet.
2
1.5
0.8
0.6
Perpendicualr Velocity Spread
-
-
Beam Pitch Factor
0.5
a
Maximum Electric Field
0.4116
19
18
17
(kV)
voltage
Mod-Anode
20
Figure 3-32: Operational parameters for the electron gun at 32 kV. Gun coil at 25%
of rated current.
86
..................
.............
1.15
2
1.5
0.95
E
0.75
'.5
Perpendicualr Velocity Spread
-
-
Beam Pitch Factor
a
Maximum Electric Field
0.55
'
22
22.5
'
1
23
23.5
24
Mod-Anode voltage (kV)
24.5
25
Figure 3-33: Operational parameters for the electron gun at 40 kV. Gun coil at 25%
of rated current.
1.2
3
2.5
E2
0.8
1.5
Velocity Spread
-- Beam Pitch Factor ax
Maximum Electric Field
0.6 -Perpendicualr
0.41
30
Figure 3-34: Operational
of rated current.
-0.5
'
'
30.5
31
31.5
32
Mod-Anode voltage (kV)
parameters
for the
electron gun at
32.5
50 k.
33
Gun coil at
25%
3.9
Conclusion
The complete design of a novel 250 GHz gyro-TWT based on a PBG interaction
circuit has been presented. This design relies on the frequency dependent distributed
loss provided by the interaction circuit to suppress both forward and backward wave
oscillations from lower order modes. Linear theory and non-linear simulations have
shown reasonable operational parameters and circuit length will provide the desired
gain and bandwidth. With 60 dB of amplification and over 3 GHz of bandwidth the
design meets the requirements of pulsed DNP experiments. An electron gun that
meets the operational requirements of the gyro-TWT has also been designed. With
~2% velocity spread from 32-50 kV while providing a an a of 0.6-1 the triode configuration provides the necessary tunability to -5%.
maximize the effectiveness of the
amplifier. Surface roughness and machining errors are expected to increase the velocity spread to A wraparound input coupler with more than 50% power transmission
from fundamental waveguide to the PBG will provide the required mode conversion
from the solid state source to the amplifier circuit. For the current configuration a
multi-step uptaper provides -20 dB reflection from cutoff to 260 GHz while allowing
for axial beam extraction.
Chapter4
Short Pulse Amplification
4.1
Introduction
In order for an amplifier to perform adequately in pulsed DNP NMR experiments it
is not only required to produce kW power levels, it is also necessary to amplify pulse
trains without distortion. For pulse trains with a frequency spectrum on the order of
the amplifier's bandwidth this could prove problematic and has not been previously
investigated. Using an existing experimental setup at 140 GHz, linear amplification
of pulses was thoroughly investigated and compared to a theory for amplification of
short pulses in a gyro-TWT that was derived in Section 2.7. In a gyro-TWT gain
occurs near the intersection of the beam line in Equation (1.1) and the waveguide
dispersion line in Equation (1.2) as shown for the HE 06 mode of a confocal waveguide
in Figure 4-1. The 140 GHz gyro-TWT used in this experiment has produced a 3-dB
gain bandwidth of over 1.5 GHz (1.1%) with a peak power of 570 W from a 38.5
kV, 2.5 A electron beam [29]. A maximum output power of 0.82 kW was achieved.
The amplifier circuit is an overmoded quasi-optical confocal waveguide consisting of
three 7 cm long gain sections separated by two 3.7 cm severs. Figure 4-1 shows a
cross section of the confocal waveguide with the waveguide rails of 6.93 mm radius
separated by a distance L1 of 6.93 mm. The gyro-TWT operates in the HE 0 6 mode.
The electron beam is tuned to a grazing intersection with the HE 0 6 waveguide mode.
The beam line intersection with the HE 05 mode can lead to backward wave oscillations
limiting the beam current in the circuit. By adjusting parameters such as the main
magnetic field, the electron beam voltage and the voltage on the control anode, the
beam line may be moved from close to the cutoff position to much farther from the
cutoff position.
180
2a =4.8 mm
HEo7
170 -
K 150-
..
....
HEos
140
L-
130-
120
-
-3000 -2000 -1000
0
1000
V 28
V= 35.4 kv
2000
3000
HEo6
Axial Propagation constant (1/m)
Figure 4-1: Dispersion diagram of HE06 gyro-TWT in a confocal waveguide (left) and
cross sectional view (right). The annular electron beam has a 4.2 mm diameter.
For this study, the gyro-TWT was first operated in longer pulses, on the microsecond scale, to establish the gain and bandwidth in long pulse (quasi-CW) operation,
prior to the short pulse tests. Results were obtained at voltages of 35.4 kV and 28
kV, with 1.8 A beam current and 5.01 T magnetic field. Figure 4-2 and Figure 4-3
show the theoretical and measured gain bandwidth for two operating conditions that
were investigated. The measured gain, for a 1 ps pulse, was as high as 30 dB and the
output power was about 10 W, which is well within the linear gain regime. The linear
regime operation was verified by measuring the input power vs. output power over a
range of input power values, as shown in Figure 4-4. Simulations of the gyro-TWT
were performed using the computer code MAGY [10] with a beam velocity ratio a
= 1.1 and 1.4 and a perpendicular velocity spread of 9.5% and 12% for 35.4 kV and
28 kV, respectively. The measured data points in Figure 4-2 and Figure 4-3 are in
40
35
....
.
. ..
....... . .. .. .. .. ..
.
.
.
.
.
. ..
.*
..
..
..
.
.
.
.
.-
30
. ... .. .. .. .. .
..
. .. .. . .. . . . . . . . . . . . . . . . . . . . I.
.
.
25
....
....
....
....-..
20
15
10
5
135.8
136.4
137.6
137
Frequency (GHz)
138.2
138.8
Figure 4-2: Measured (dots) and calculated (solid line) gain-bandwidth curve for a
beam voltage of 28 kV.
30
..
20
10
136.6
137.2
139
137.8 138.4
Frequency (GHz)
139.6
140.2
Figure 4-3: Measured (dots) and calculated (solid line) gain-bandwidth curve for a
beam voltage of 35.4 kV.
0.7
0.6
0.50.4-
Q0
0.3 0.2
0.10
0
0.5
1
1.5
P.Pin (W
(mW)
2
2.5
3
Figure 4-4: Measured input and output power at 137.8 GHz for the 35.4 kV operating
point. The gain is 23.5±0.34 dB
reasonable agreement with theoretical predictions. The 3-dB bandwidths of both
cases are about 1.2 GHz, with a center frequency of 137.4 GHz and 138.1 GHz for
the 28kV and 35.4 kV operating points respectively. Following the microsecond gain
experiments, we conducted tests of the gain and dispersion of the gyro-TWT with
picosecond input pulses.
4.2
Experimental Setup
Figure 4-5 shows a schematic of the picosecond pulse amplifier experiments. The
picosecond drive pulses are generated by a solid-state source consisting of an Xband mixer switch with an up-converter to 35 GHz and a Virginia Diodes, Inc x
4 multiplier to 140 GHz [30].
The driver input at the gyro-TWT after the 3-dB
directional coupler as shown in Figure 4-5 was approximately 16 mW over the 135-143
GHz range in pulses as short as 400 ps. The detector diode outputs were measured
by an oscilloscope with a 10-GHz bandwidth and a 40 GS/s sampling rate. The
.
::
..
............
:::::::..::
'..::::::
:..:::::::::::
...........
..........
. .....
__ . ..
- ......
relative temporal responses of the input diode and the output diode were tested and
compared while simultaneously connected to a common directional coupler. The
temporal responses of the two detector diodes from 400 ps to 1 ns agreed to within
5%.
Isolator
3-dB
Attenuator coupler
Quasioptical n aq
(input)
Solid-statenptOclooe
driver
gyro-TWT.io
ipi
(output)
IptOclocp
DiodeL
t
Sub-ns Function
Generator Gate
Figure 4-5: Schematic diagram of the picosecond pulse amplification in a quasioptical
gyro-TWT.
4.3
Short Pulse Amplification
As described in Section 2.7, the pulse in the interaction circuit is expected to suffer
from two dispersive effects. Both waveguide dispersion and gain dispersion will play
a major role in the broadening of pulses in the amplifier. If the amplifier is operating
in the linear regime, the pulse's interaction with the electron beam can be described
by
aza(z, w) =
-ikza + G(w)a
(4.1)
where a(z, w) is the frequency spectrum of the pulse, k2 is the axial wave number
and G(w), the linear gain over the spectral range of the pulse, can be determined
from theory or from experiment [16], as in Figure 4-2 and Figure 4-3. The effects of
dispersion can be modeled as distortions to the frequency spectrum of the pulse as it
travels through the interaction circuit. Theoretical broadening curves were generated
by observing the broadening of Gaussian pulses as they traveled through the circuit.
For a pulse A(zo, t) at the start of the circuit, the frequency spectrum is given by,
a(w) =
I
A(zo, t)e-'w dt.
(4.2)
and we can propagate the pulse in the frequency domain using the solution to Equation (4.1)
a'(z, w) = a(w)eizze()z
(43)
For the Fourier transform of a pulse, if we amplify and advance the phase of every
spectral component, it will be equivalent to propagating the pulse through the circuit.
Performing an inverse Fourier transform on this new spectrum
A'(zi, t)
=
j
a(w)e-ikzz eG(w)zleiWt dw
(4.4)
produces the output pulse in the time domain. Equation (4.4) shows that gain dispersion plays a significant role in broadening of picosecond pulses when the spectral
bandwidth of the amplified pulse is comparable to the gain bandwidth of the gyroTWT. In addition to measuring G(w) experimentally, the cutoff frequency must be
determined in order to calculate the phase advance produced by kz. Using a network
analyzer to measure the cutoff frequency for traditional fundamental mode waveguide is relatively easy. However, the interaction circuit has the added complications
of being highly lossy and requiring coupling from a fundamental mode waveguide to
the HE 0 6 mode. Furthermore, the cutoff frequency could be altered when the circuit
is installed in the magnet bore. An alternate approach to determine the cutoff frequency is finding the best fit for the time of flight for a pulse traveling through the
circuit.
In order to benchmark our procedures, we measured the dispersion of picosecond
pulses in an empty WR4 waveguide with cutoff frequency of 139.25±0.1 GHz. The
cutoff frequency was measured using a vector network analyzer (VNA) as shown in
...
..........
Figure 4-6. It is simple to calculate (blue line) the attenuation of the TE10 mode in
10 cm of WR4 waveguide for a specific cutoff frequency, showing excellent agreement
with the measured data (green line) in Figure 4-6. It is important to calibrate the
measurement at the input and output ports of the WR4 waveguide. The inclusion of
the WR4-WR6 transitions in the measurement (red line) shows a significant difference
in the attenuation due to the characteristics of the transitions.
0
-5-10E
S-15-20
-25
138
140
142
144
Freq (GHz)
146
148
150
Figure 4-6: Attenuation as a function of frequency in a 10 cm section of WR4 waveguide.
In the case of empty WR4 waveguide only waveguide dispersion, as given by
Equation (1.2), is present. In Figure 4-7(a) pulses traveling through 10 cm of the
WR4 waveguide showed pulse transit time delays that are highly dependent on the
cutoff frequency of the waveguide. The time of flight data in Figure 4-7(a) is in
excellent agreement with the 139.25 GHz cutoff measured using the VNA.
A waveguide filled with an active medium (the electron beam in this case) shows
an additional dispersive effect due to that medium, especially near cyclotron resonance. In Figure 4-7(b) we show the measured transit time delay for 500 ps pulses
2.55
6.35
Experiment
......................
.--Calculation
.................................. 5.852.25
I
..................
..........................
1.95
5.35
............
..............
.........
........... ..........
......... 4.85 (1. )
1.65
..........
......... 4.35
.
..
.
. ...
.
.
.
.
..
1.35
140
141
142
143
144
145
136
(1)Empty WG
(11)
V= 28 kV
. ((11) V= 35.4 kV
137.2
138.4
Frequency (GHz)
(b)
Frequency (GHz)
(a)
139.6
Figure 4-7: Measured (dots) and calculated (solid line) transit time delay of 500 ps
pulses in (a) 10 cm long WR4 waveguide with a cutoff frequency of 139.25 GHz and
(b) the 28.4 cm gyro-TWT confocal waveguide structure with a cutoff frequency of
135.0 ± 0.3 GHz.
1000
*
900 -
Experiment
WG Dispersion
800 F
700
600
500 r
400'
140
141
142
143
Input Pulse Freguencv (GHz)
144
145
Figure 4-8: Output pulse width as a function of input frequency for 10 cm WR4
waveguide and a constant input pulse width of 500±25 ps.
96
traveling through the 28.4 cm confocal waveguide structure of the gyro-TWT at two
different voltages as a function of frequency. Equation (4.4) was used to calculate
the theoretical results shown in Figure 4-7(b). The theory curves use the G(w) that
was measured in Figure 4-2 and Figure 4-3 and a cutoff of 135 GHz. These results
are in excellent agreement with the experimental data at the two different operating
voltages. In Figure 4-7(b), (I) is the calculated transit time delay in the gyro-TWT
including only waveguide dispersion, that is, for the empty waveguide with no electron beam present; (II) and (III) are the measured and calculated transit time delay
of picosecond pulses in the gyro-TWT for two different beam voltages, namely 28 kV
and 35.4 kV. These results show that the active medium, the electron beam, has a
significant effect on the transit time.
The broadening of ps pulses in cold WR4 waveguide was also used as a bench
mark.
The waveguide dispersion causes the ps pulses in the WR4 waveguide to
undergo significant pulse broadening. The broadening results for a constant input
pulse width of 500i25 ps as a function of frequency are shown in Figure 4-8. As
expected, pulses of a frequency that is closer to the waveguide cutoff are broadened
much more severely.
Similarly, Gaussian pulses of the same width but different frequencies can be amplified in the gyro-TWT where both dispersive mechanisms will be present. For the
28 kV operating point pulses with an input width of 425±25 ps were amplified for
a variety of input frequencies.
The results in Figure 4-9 show that the minimum
broadening occurs slightly above center frequency of the gain spectrum because of
the strong effects from waveguide dispersion. As the frequency moves away from this
minimum, either closer to or further from the cutoff of the interaction circuit, the
broadening increases. The frequency spectrum of a pulse that is amplified with an
input frequency close to the edge of the gain spectrum is severely distorted resulting in pulse broadening. There is very good agreement between the measured and
calculated broadening values when both forms of dispersion are included. Figure 49 also shows the theoretical broadening curve for the interaction circuit with only
-........
...
11
............
...
....
..
- ----.
...
. ......................
..........
.....
.....
..
1200
.
*
Experiment
Waveguide Dispersion
Waveguide and Gain Dispersion
1100 I--,
tA
CL
900
800
0
700
138.4
138
137.6
Input Pulse Frequency (GHz)
137 .2
136.8
138.8
Figure 4-9: Output pulse width as a function of input frequency for a beam voltage
of 28kV and a fixed input pulse width of 425±25 ps.
I
1
'I
-+- Input (Exp.)
I,
-- Output (Exp.)
Output (Cal.)
0.9
O 0.8
0 0.7
0.6
0.5
0.4
0.3
0 0.2
z
0.1
3804
1
0.9
-0-Input (Exp.)
-0-Output (Exp.)
Output (Cal.)
0.8
0.7
I I
0.6
0.5
0.3
0.2
0.1 A6
9,
3806
3808
3810
Time (ns)
3812
3814
3804 3806
3808
3810
3812
3814
Time (ns)
(b)
Figure 4-10: Pulse shapes for an input pulse of 580 ps at the 35.4 kV operating point.
The output pulse at (a) 137.3 GHz has a measured width of 1045 ps and (b) at 138.13
GHz has a measured width of 660 ps.
waveguide dispersion. It is evident that the broadening data is in stark contrast with
these calculations. This shows the importance of considering not only the effect that
waveguide dispersion will have on short pulse amplification, but also the effect of gain
dispersion.
The time domain recordings of pulse envelopes taken by detector diodes, as shown
in the schematic of Figure 4-5 before and after amplification in the gyro-TWT, can
also be used to compare with theory. Figure 4-10 shows an example of the measured
temporal shapes of input and output pulses. Operating at 35.4 kV with a 600 ps input
pulse, the detected output pulse envelopes vary with frequency in good agreement
with the numerical simulation results based on Equation (4.4). From Figure 4-10, we
can clearly see that for a sub-nanosecond pulse traversing an interaction space with a
gain bandwidth of 1.2 GHz, the output pulse will suffer significant pulse broadening
in the time domain, especially if the frequency of the pulse is not centered in the gain
spectrum.
4.3.1
Pulse Broadening Curves
Figure 4-11 and Figure 4-12 show experimental results and theoretical calculations
of the output pulse width vs. input pulse width for the gyro-TWT, showing clear
evidence of pulse broadening effects. For very long pulses (not shown in the figure),
greater than a few nanoseconds, there is no pulse broadening, as expected; the output pulse has the same width as the input pulse. Results are presented for the two
beam voltages for which the gain spectra are shown in Figure 4-2 and Figure 4-3.
Figure 4-11(b) and Figure 4-12(b) show pulse broadening curves at the center of the
gain spectrum where the broadening due to gain narrowing should be minimized. We
also find that even though both operating points have a similar gain bandwidth, for
the case of Figure 4-11(b) the broadening is more severe than Figure 4-12(b). Theory
shows that this arises because the frequency of the pulse in case Figure 4-11(b) is
0.75 GHz closer to cutoff, thus increasing the waveguide dispersion contribution to
1400
1400
*
1200
--
-
1200 --
Experiment
WG-BW Dispersion
1000oo
10000
400
1400 - -
200
0
200
200
Experiment
---WG-BW Dispersion
.
800
1000
400
600
Input Pulse Width (ps)
1200
-.-.-.-.-.-
0
1400
200
600
800
1000
400
Input Pulse Width (ps)
1200
1400
1200
1400
(b)
(a)
0
0
800.-..---.-
...............
600-
--
-
600
800
......
400-
400200
0
---
0
200
200
Experiment
WG-BW Dispersion
800
1000
400
600
Input Pulse Width (ps)
1200
1400
(c)
0
200
1000
600
800
400
Input Pulse Width (ps)
(d)
Figure 4-11: Measured (dots) and calculated (solid lines) output pulse width after
amplification as a function of input pulse width for 28 kV and (a) 136.94 GHz, (b) 137.4
GHz, (c) 137.92 GHz and (d)138.2 GHz.
100
1400
1400
1200
1200
1000
1000
800
800
600
600
400
400
-c
0.
4
0
*
200
_
0
200
400
600
1000
800
0
1400
1200
200
400
600
Experiment
WG-BW Dispersion
800
1000
1200
1400
Input Pulse Width (ps)
Input Pulse Width (ps)
(a)
(b)
1400
1200
0
1000
800
600
0
400
0
200
Experiment
WG-BW Dispersion
0
200
400
600
800
1000
1200
1400
Input Pulse Width (ps)
(c)
Figure 4-12: Measured (dots) and calculated (solid lines) output pulse width after amplification as a function of input pulse width for 35.4 kV and (a)137.3 GHz, (b)138.13
GHz and (c)138.8 GHz.
101
the broadening. Figure 4-12(a)-(c) was produced for the same beam voltage (35 kV),
but for different input pulse frequencies. Figure 4-12(b) displays the least broadening
because it is centered in the gain spectrum of the amplifier at 138.13 GHz. Figure 412(a) is both 0.85 GHz below the center of the gain spectrum and closer to cutoff.
As a result, pulse broadening arises from both gain narrowing and the proximity to
waveguide cutoff, resulting in very large broadening. At the 138.8 GHz operating frequency of Figure 4-12(c), pulse broadening is also worse than Figure 4-12(b) because
this point is 0.65 GHz above the center of the gain spectrum, causing gain narrowing.
Data taken at 28kV is shown in Figure 4-11(a)-(d) for different input pulse frequencies. Similar observations can be made about pulse broadening. The closer to cutoff
the more severe the effect of waveguide dispersion. So much so that the minimum
broadening is not in the middle of the gain spectrum (also visible in Figure 4-9).
Furthermore, amplification of pulses at the limits of the gain spectrum also produces
severe pulse broadening. Theoretical calculations of output pulse width vs. input
pulse width were carried out for the cases shown in Figure 4-11 and Figure 4-12 using Gaussian pules and Equation (4.4), based on the experimentally measured gain
spectra of Figure 4-2 and Figure 4-3, respectively. As seen in Figure 4-12 and Figure 4-11, the theoretical curves are in very good agreement with the experimental
measurements.
4.4
Summary
In summary, the amplification of a sub-ns pulse in a 140 GHz gyro-TWT was successfully studied over a wide range of parameters. The experimental data of pulse
broadening and transit time delay in the gyro-TWT are in good agreement with calculations in the linear gain regime that take into account the effects of waveguide
dispersion and finite gain bandwidth. These results show that amplification of short
pulses in a gyro-TWT will display significant broadening if the operating point is
selected close to cutoff or close to the edge of the gain bandwidth. Based on the
present results, it is crucial to both choose an operating frequency at least several
102
percent above the cutoff of the waveguide circuit and to provide the necessary gain
bandwidth of at least 1 GHz. These guidelines will permit use of the gyro-TWT for
its intended application, namely amplification of pulses down to 1 ns pulse length
for use in pulsed dynamic nuclear polarization enhanced nuclear magnetic resonance
experiments. The criteria described here are satisfied with the design of the 250 GHz
gyro-TWT presented in the previous chapter. The 250 GHz gyro-TWT was designed
to have 3GHz of bandwidth centered at 3% above cutoff.
103
104
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